I get mine EVERY time I'm in WDW without fail. It really stinks! I'm very irregular and will usually only get 3 to 5 periods a year. You're probably thinking "lucky girl!" right? NO!! When I do get them, they are horrible. They make up for the ones I missed -- very heavy and once I had a period that lasted for 10 weeks. Not fun!! Even on the pill I'd still have bad cramps and it didn't regulate it too much.
I tried Nuvaring which worked well for a few months (I kept it in an extra week every other month and changed right to a new one, so I would skip every other period). Then, RIGHT BEFORE I went to WDW -- I got my period while still wearing the Nuvaring! Figures. But if it works for you, it takes very little getting used to and is super easy to use, very hassle-free and convenient. I'd give it a try if I were you.
In the last 2 years, I went to WDW 6 times, and had 6 periods. Guess when?

I've learned to just live with it. Unfortunately, due to other health issues I am already on pretty heavy-duty narcotics and can't take any OTC pain relievers with them (and they don't touch the cramping) so I can't help much on that question but I just wanted to say that I can totally sympathize. OH -- and
bring plenty of supplies!! I found out the hard way that pads/tampons are VERY expensive in WDW!!
I have heard that staggering Tylenol with Motrin works wonders for moderate to severe pain, and I wholeheartedly agree with staying ahead of the pain. Stay on a schedule and take your next dose before you need it, or it won't work as well. You have to be smarter than your pain!