I'm a native speaker, not a grammar expert; hopefully others will chime in. I don't think any of these are natural. OBLIGATORILY sounds very formal. Actually, I've never heard the word used. Although I'm not sure your use of it is wrong, I think most native speakers would be confused by these sentences. NECESSARILY: I had to think about this one. Again, I'm not sure your sentences are wrong, but they sound unnatural. I realized that NECESSARILY is almost always used in the negative. Ex: Just because she is beautiful doesn't necessarily mean that she isn't also intelligent. Just because I bought a nice shirt, it doesn't necessarily mean that I am rich.

I agree with Dan. If I needed to express obligation I might say, ‘It is obligatory that’ or ‘I have an obligation to’, never ‘obligatorily’. ‘Necessarily’ is used a lot, but usually in a way that hides its literal meaning. Are you going to grad school next year? Not necessarily. (Just means ‘maybe not’) Of course, you could use ‘necessarily’ with its literal meaning, in writing or something formal. Climbers’ packs are necessarily much heavier than hikers’. They have to carry ropes, crampons, helmets etc.

This sounds like the word обязательно, but in English we prefer to use words like 'need' or 'should'. I try to translate the same way, but (Byelo) Russians express ideas with different words. 1. Please take yesterday's documents with you. If you want to insist, begin it with "Please do take...." 2. I will need to draw up this contract tomorrow / I will need to engross this contract and it'll be ready tomorrow. To engross = to draw up, to write (legal term). 3. ...you definitely should wear it/ put it on for today's meeting. 4. ... He should see the doctor./ He definitely should see the doctor. / He really should see the doctor.