I know nike has some great sports underwear. I have their compression shorts and their normal boyshort underwear (wear them under my tennis dresses). Very breathable and they do come in white. I imagine they would work nicely for the white gi's. Mine isn't bleached though, so plain white would be blazing through my gi pants. You could always give those a shot! As for your pants during testing... hmm... you could always go with David's idea of duct tape! Use a pretty color though.
A quick question about your drawstrings, do they have knots at the end? If not, try tying knots at the end of each string. Maybe that will help.