After the paper has soaked (I leave it submerged over night) do I get most of the water off of the paper before I start sanding or should it still be dripping wet?
You dont have to let the paper soak overnight. You just have to dip it in water, leave it wet, and sand untill you need more water.
it doesn't need to be soaking. Neither do you need it soaked all night, just use some more sanding paper
My Gear:

Schecter Hellraiser C-1 FR (EMG 85/81)
Mesa Boogie Stiletto Deuce
LOPO Custom 2x12 (Celestion V30s)
Actually, for finer and smoother sanding, it is advisable to leave it to soak over night...

As for your issue with water, when you sanding between coats and doing the final wet sand, use copious amount of water... so yea, leave it really dripping... This helps to get a smoother surface and a deeper shine...

For wet sanding I either:
A: use a spray bottle with soap and water and just spray a little bit at a time
B: use a bucket of soap and water, dip a paper towel in it, wipe over the area im sanding
Yeah, use lsot of water, if you can't slide across hte body smoothly then you probably don't have enough water.
Ok, so it's normal for the paint to come off a little onto the paper, hands, surrounding work area, clothes, etc. . Alright, thanks again guys!