Hi, Brody-san
I don't fit the profile, I'm not fluent in a second language.
But I hope you don't mind my 2 ¢ worth I can't resist (you can always scroll past)...
I think it's like driving a car, you can't think about it. You just DO it. When you learn you're very conscious of what you're doing though. When does the switch happen? How does it happen?
I think language is the same way. You haven't the time to do the translating thing. If you do it gets in the way. It just happens. How to get to that level? I think practice and repetition and rehearsal. I always fall back on my stock phrases when talking. Through rehearsal I expand these phrases and add new words to the mix. I sometimes write out scripts. Making an oral presentation is great as it's one sided! but it builds lots of confidence. If it opens onto questions the topic is fairly well defined and you can usually pull out an answer from something you've already said.
(the way you bring out foreign words I believe is called "mustering". You might be interested in the Yomiuri Shibuns language connection. It deals with teaching English but they have interesting articles.
http://www.yomiuri.co.jp/dy/features/language/)
I think rehearsal happens in English as well. You plan what you might say at an interview or presentation (or a first date!) maybe in your head and not out loud, but I doubt I'm unique in doing this.
We are so used to hearing actors. But their lines are scripted rehearsed edited done again and again. Real people don't talk that way. Try listening to people talk in English. If they are anything like people in UK they will constantly repeat themselves and each other, use a fair deal of padding and use (NOT have) a limited amount of words.
But maybe become an actor in Japanese, play the role of gaijin speaking Japanese. Rehearse it. Just an idea. It's the sort of thing shy people are told to do. (did you make a Nanase-san video?)
And you have to loosen up a little. (I know I do!) Make mistakes; the idea can still get through. Very hard to do for someone fluent in English and well educated but you also have to make do with simpler language. And when you listen you may never get 100% but you need strategies to pick out what you need. You're not going to pass for native and I'm not sure they care. And communication is 2 way; you can rely on (hope for) the native speaker to help you out.
I know all this from my own learning experiences and from dealing with a lot of people whose English ranges from "how did they let them study here" to "have to listen hard to know it's a second language".
On meanings of words and phrases. The more I learn the more I find that there isn't a match between English and Japanese. How they phrase things, what specific words refer to, why this word not that one, (to me スポットライト and 電光 mean the same thing, spotlight. But I would tend to favour 電光 because it's nice and foreign-sounding! But it actually sounds very archaic to my teacher! )
If there was a simple match machine translations would work, but they don't. Ultimately you have to just let go and become Zen-like or something

I think to reach that you'll need to use word lists and English (there's some sort of critical-mass you have to reach by this method) but eventually you'll have to trust yourself to just DO it. And it happens... I'm sure of it.
I don't translate things like 本を買いました。It does go in and out without thinking as a direct mapping to the concepts book and buy. I can do this in Irish as well (ceannaigh me leabhair). But not French which I've mostly forgotten (je vende un livre, I sell a book?-- I can't even remember past tense any more! use it or lose it!) But once I think I had to translate and with more complex concepts I
still have to work things out. But eventually through practice I think I'll get there. (unfortunately I don't get or take enough opportunities to practice) I notice more and more that I'm understanding the Japanese around me (TV, people speaking,songs, jPod podcasts). It once took me days to read an email now somewhat less as I'm used to the writer. (oddly I find it much harder when I'm being spoken to directly, stagefright maybe)
As you have mentioned in your post, conversation is a come as you are experience!
I'm very interested in how you get on removing English from your studies. Once you have a base ability I'm sure an immersive experience does wonders and probably accelerates learning. (It must be exhausting to have to deal with a foreign language 24/7 though)
How to remove English.
The classroom method that is used on me is teaching Japanese through Japanese! Yes we need English but it's less and less of the time, I can now
attempt to talk about the grammar points etc. in Japanese as well. The rule is to not speak English until the English 20 mins or less at the end of class. It was a shock the first day. Chika-san arrived and just talked in Japanese at us and used pictures and mime and got us to repeat things. I think everyone just thought OMG she doesn't speak English how are we going to learn anything!? But we did and I think by removing English from day 1 we might have learnt things better. We also tend not to translate as a class activity or even as a homework. It's more here are grammar points here is vocabulary now put them together and talk about something real that you know about. (of course this is not a university exam course and is very slow at 1 day a week and there's no way I could even sit in on a middle school lesson in Japan!!)
On specialist vocab. like legal terms or medical terms or film terms. You need it if you need it. BUT if you do need you probably need a large general vocabulary and ability in order to back it up. So you have to do the crawl before walking thing. And for my money can I have a beer and some sushi is going to be of more immediate daily use than Stick a 1K on that lowboy set it English with a quarter CTB and put a cookie on a c-stand in front of it.

(it's English really! but my specialist language, someday I might be able to say it in Japanese)
Of course it would be great if you could learn each individual word in context, the same way you did when you were learning English as a kid. But that's just not feasible as an adult.
I disagree. You might not have the
time but you have a better ability and intellect to learn words from context. Again after reaching whatever the critical mass is. And as a skill in itself "figuring out" is more useful than "memorise list", I think. And compared to English with kanji it is possible to derive meanings from the symbols themselves with a bit of practice and effort. I can't recall ever using my knowledge of french or latin to figure out English vocab but I have figured out unknown kanji compounds.
But in the end use
every tool that seems to work for you. If you find something works better than others for you concentrate on that maybe. Relax, have fun, accept your current ability, work to improve it but keep perspective.
I envy you (both) going to live in Japan. I'm sure you'll have a blast. 頑張ってください!!
(oh dear another long post hopefully not BS; it must be because I'm Irish, gift of the gab and all that

)