My experience with Russian is that it simply has too much grammar to drill and too much vocabulary to learn to do it all "up front". Grammar translation approach + anki for Russian is how I failed to learn Russian on my first attempt. Grammar translation is the method that is commonly recommended for Russian though and pretty much all the textbooks and courses I have encountered, except Assimil, were very heavy on drilling. This makes total sense, because Russian is a difficult language and it takes an awful lot of drilling if you want to speak it correctly. All that grammar translation aims for correct production though, which is not your aim, so you should choose a different approach that doesn't waste your time with drilling.
All you need is a quick overview over all the grammar. The courses I used for this kind of overview were Assimil and Duolingo. I did the whole Duolingo course, but it's only useful up to the last checkpoint. About halfway through was enough to start reading with a dictionary. I abandoned Assimil about halfway through, because reading was teaching me more at that point. It's a good way to get started though, especially if you shadow the audio.
Speaking of audio: Although you don't intend to speak or listen to it, I would recommend that you familiarise yourself with the pronunciation anyway. If you ever end up in the situation where you have to ask for a Russian title in a bookshop or refer to a book in Russian it would be rather uncomfortable not to know how to pronounce it correctly. The problem is that Russian pronunciation is not entirely predictable from the written form, because word stress is not marked. You will have to listen to the words to know how they are pronounced. That might seem like too much work if you never want to speak the language, but I find that listening to the word helps with remembering it too. It also makes sure that I don't misspell the cyrillic when I write the word down.
How to get familiarised with the pronunciation? One part of the pronunciation is obviously learning the alphabet and how to pronounce the letters correctly. For this I found the course Russian World 1 on youtube most helpful. It goes over everything very slowly and they even teach you the cursive handwriting. No matter whether you ever write by hand, it's actually a useful exercise, because you will see cursive in books, especially in non-fiction books, and on websites. After that using minimal pair drills is the most effective to be able to distinguish all the different ш, щ, ч sounds. If you can't distinguish them, you won't be able to pronounce them correctly, so this is really important. One source of minimal pair drills would be Gabriel Wyner's pronunciation trainer, but you can also find them in the audio material of the audiolingual course Modern Russian - the audio can be found online for free. I used both, but found Wyner's trainer better. It runs on anki and his pronunciation courses are the only reason why I keep anki installed. I generally hate flashcards and don't use them at all anymore beyond these minimal pair drills. To really get the pronunciation straight, shadowing half of Assimil helped me a lot too. It helps to internalise grammar too, which makes it easier to understand, so it's doubly helpful. After that, simply make it a habit to listen to every new word you encounter, it only takes a second and it really helps to remember.
If you follow this strategy you might even be able to listen to Russian without ever having made it a priority. For French I followed the same strategy and I only needed to watch one season of a dubbed series with subtitles to then be able to understand spoken French. And with French I didn't even listen to every new word, because French pronunciation is predictable from the written form. My own *correct* subvocalisation while reading 5000 pages was enough to train my ear for understanding spoken French. Basically, if you focus on the pronunciation up front, you get listening skills almost for free. Although you might not want to watch TV there are actually an awful lot of interesting talks and lectures out there in Russian, so putting in that little extra effort for pronunciation might one day come in handy.
So, that's basically my method, I use a pronunciation trainer, Duolingo, and shadow Assimil while I already start to read graded readers with Learning with Texts, as soon as I can parse the grammar of simple sentences. For me reading is also the most important skill, so I start with it as early as possible. Once I get bored with graded readers - and that happens very quickly -, I pick a series of books to read and start on my first 5000 pages of real content. Russian sentences can be long and complicated thanks to the strange word order and crazy grammar, so in the beginning I would definitely choose a translation from English so that the sentences don't get too wild. It also helps to choose something for which you can access an English version in case you run into a sentence that makes no sense even when you know all the words and grammar. At the beginning there is usually one on every page even in translations. In originally Russian writing it will be more like 2-5. I also choose a series of books because then the vocabulary used will be on always the same topic. Since there is an awful lot of vocabulary to learn in Russian it helps to be somewhat systematic. If you can stand them, then young adult novels are the best choice. I picked a fantasy series for Russian and the writing style tends to be somewhat erratic, somewhat old-fashioned or stilted at times, so it wasn't the best choice. For French I found Harry Potter to be a good choice because the language also becomes more complicated over the course of the books. It was a bit of an easier start than I had with the erratic stilted style of the fantasy series in Russian.
For vocabulary I don't use any additional methods, simply reading with a dictionary - i.e. Learning with Texts. Typing in every word definition into the database seems to be enough to make the words stick eventually. The advantage is that I only encounter vocabulary that I actually need and it will always be in context. Words that appear often I will remember, the others will simply slip away again and that's totally fine. And I get to read real content for native speakers very early. It's challenging in Russian, but for me it definitely works. And if reading is your goal, then I would suggest to start with it as early as possible, because it's considered one of the most difficult skills when learning Russian. By the way, with LWT it also seems to help my memory to actually write down the Russian word as well, so I put the Russian word in the Romanisation field. This would be tedious if I didn't know how to touch type in cyrillic, so I recommend learning that too. It's probably the single most useful skill when learning Russian, because it helps to make dictionary lookups a lot quicker than if you have to deal with transliteration tools or hunting for the right keys on an unfamiliar keyboard layout. You can learn it here:
http://www.keybr.com/Another tool I can recommend, especially if you seem to have problems parsing Russian sentences, is Clozemaster. It exposes you to a lot of natural sentences and vocabulary, so it's good additional practice for the beginning. Once you have less problems with long sentences in books it won't be so useful anymore, but up to that point it's very helpful.
And the single most important piece of advice when learning Russian: Be prepared for this to be a long and somewhat tedious journey. Russian is a difficult language, difficult to understand and even more difficult to produce. Oodles of grammar, seemingly endless mountains of vocabulary. Even after having read a whole 250 page book with LWT I still run into pages with 30% new words or sentences I simply can't parse. This early intensive reading approach is also taking me more than double the time than it took me for French, because there is just so much more vocabulary to learn. That's just what Russian is like, so don't get disheartened by it.
In any case, I hope these tips are helpful to you and good luck on your Russian journey!
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French was putting up a lot less resistance and it was more "fun", because I already knew English and Spanish. I simply had to struggle less, because I got 70% of it for free. Even so, with the intensive reading approach I'm making pretty steady progress with Russian and I see this progress reflected when I encounter simpler non-literary texts. I'm also not lost on Russian websites anymore and I understand the occasional simple conversation on the street. And the method is a walk in the park in comparison to the anki grammar torture I tried before! I was thoroughly overwhelmed with that approach, because it seemed like such a mountain to climb - all that grammar drilling - just to get to a point where I would still need a dictionary to read?! What a waste of time! It was really no wonder that I gave up, because I simply didn't see any improvement in my comprehension. After 3 months of grammar + anki torture children's books were just as indecipherable as on day 1. Even with a dictionary I couldn't make sense of them! Assimil + Duolingo + intensive reading is definitely the better method for tackling comprehension and I've minimised the torture part as much as possible. Doesn't change that Russian grammar and word order is mad and that there are way too many words to learn though 
