I still don't know if I would need Polish residency for obtaining a russian Visa in Krakow or Warsaw.
Yes. If you aren't registered as a long term temporary resident (or better) - then you can't obtain a Russian visa in Poland.
I have about three weeks left right now before moving to Moscow.
You're cutting it fine!
You'll have to deal with it through Berlin - the best hope in this case is to get one of the agencies there to deal with it.