I do a lot of sweep picking in my playing, my advice is to sweep off the neck pickup, roll the gain/distortion back to about 6 or 7 on the dial on the amp and pay extra careful attention to muting the strings with your left and right hand. The only string that should not be muted is the string being picked, as soon as you 'sweep' to the next string the previous one should be muted (you also need to mute the proceding strings - that's where both hands come into play). It's the most difficult of all to master, took me about a year of trying and getting nowhere then one day the penny dropped and I 'got it'. Stick with it, it takes a while!

PS the reason it sounds good clean and muddy as hell when distorted is because you are not dampening the strings - too much ringing out that only gets picked up when you boost the gain. I'd keep the volume knob at the max setting, get it sounding good off the neck pickup then try if off the bridge - it's a lot more difficult especially if you use hot pickups. Alllof the big sweep pickers like Rusty Cooley, Michael Angelo, Jason Becker, Vinnie Moore etc do a lot of flipping between pickup ssettings - the neck is almost always employed for sweep picked sections whereas the neck is for the more gainy screaming lead parts.

Hope that helps...