it's not that bad to have your period when you're at disney....it's actaully pretty alright,
because you forget about the cramps and stuff because you are so excited and having so much fun (at least for me that is true!)
if you don't want to go on brith control, i say your only option is to bring LOTS of advil/midol/whatever you take for cramps.
also...do you use tampons? because if you don't, i would recommend trying them out before your trip, they are much more comfortable to walk around in than pads.