I have an Orange halfstack. I've recently noticed something about my playing now that I've gotten it and began practicing with the band, and they've noticed too.

I rest my hand on the bridge when I play, so when I move it to pick, you hear the notes I'm picking but also the awful noise of my hand against the strings. This is seriously becoming a huge problem. Does anyone know any ways I could get this out of my playing? We're going to record soon, and I'm getting nervous.
Play quietly... then enhance the sound.

Or just buy a new half stack.

Edit: You could also try supporting your picking hand with your pinky, or turn down the tone on your guit.
Schecter Hellraiser Solo-6 FR Limited
Parker Mojo Fly
Ibanez SZR 720
Tanglewood Evolution
LaPatrie Etude DEMO
'66 Hagstrom Viking I (customized)
SGR C-7 (defretted)
Agile Intrepid 828

Amp, Pedals:
Laney LV300
i tend to stick out my pinky to suport my hand and stop it touching the bridge.
try to curl up all the fingers you dont use to pic with into your palm.

might be worth a try
Alter your technique, obviously. I've never known anyone else have the same problem, I use my right hand constantly to mute the strings I'm not playing, and don't think I've ever heard it through my amp.
Maybe focus on NOT doing that....

Seriously, it sounds like a lazy habit that you might have developed, so again, pay special attention on not doing that.

When your resting you could just drop your arm to the side, or do some huge Pete Townshend swings, or give everyone the finger. Use pauses in your playing to reach for your beer/pop, you could rest your hand/wrist on the top of the guitar. You could also try resting the tips of your ring and pinky finger on the scratch guard, or where one might be on some guitars, just below your strings.

Like the saying plainly goes, "if what your doing doesn't work, then do something else".
i rest the inside of my forearm on the guitars body and touch my pinky to the body too for support unless im strumming crazy
I read your post a little more carefully, so if I understand clearly, what you're doing is moving from resting your hand on the bridge while not playing, then sliding up the strings to the playing position. Then are you keeping your hand on the strings while playing (muting)??, or your actually moving your hand around so much on the strings while playing. If this is basically what your doing, then I specifically suggest what I mentioned above and plant your ring and pinky finger tips to the scratch guard and force your hand ato stay above the strings, and make sure your other fingers are curled up, or use the other 3 fingers to hold the pick. With enough practice you should brake your habit.