As to whether to take paper off before/after joining blocks....my opinion....depends on if you are ending up with bias edges on your blocks. If not, there is no need to keep the paper on to join your blocks. If you do have bias edges, keeping the paper on will certainly help.
Opening seams flat will also help tremendously with the bulk. Or if that is not possible, as tedious as it is, go ever so slowly when going over your joining seams - like one stitch at a time - to keep things from shifting as much as possible. Perhaps ease the pressure of your presser foot if that is an option also. Might make things wiggle less.
Good luck. And remember, by the time you get it quilted, no one is going to notice those seams not matching perfectly - not even you!