I had the impression you were reacting from repeatedly scrubbing with the standard scrubs. Just pay attention to what the options are at your hospital and try all of them....find what works for you.
You are supposed to do a full scrub to get dirt out from under your nails and stuff.
Some people I know will scrub with one of the standard agents, then will immediately wash with regular soap/water (maybe with a moisturizer) to get the scrub agent off their skin before then using the alcohol based stuff for the first case of the day. This way, they have done the mechanical cleansing they recommend. Depending on what kind of reaction you have, this may minimize time of contact on your skin and decrease your skin reaction. Worth a shot. Works with betadine scrub, at least.
If you really react badly and have no alternatives, you probably can wash with other soap and water, clean under your nails, etc., dry off, and then use the no-scrub stuff. The no-scrub reps IIRC will tell you that studies have shown this to also be effective, but it's not the official recommendation for OR use. One of my old-school attendings did this due to developing bad reactions to scrub soaps after 35 years of surgery. As a student, though, I wouldn't do this unless you get the ok from someone; you'd probably need a medical note from your derm or something to get this accommodation from your clerkship director.