I go to Disney World with my family almost every year, so I'll see what I can think of.
For the trip down, just try to keep things as simple as possible. Most GPS's have some sort of "find restaurant" or gas station feature, and most fast-food type places will have a bathroom. Worst comes to worst, pack some toilet paper in the car and try to find somewhere to pull off to the side of the road. Eat easy a day or two before, and maybe even pack food instead of eating in a rest stop. Sometimes I'll pop an Immodium or two the morning of the trip.
In the parks, there's an app called "My Disney Experience" that just came out, I believe. If you're staying on-property, you can actually make Fastpass and dining reservations through the app 60 days before. Otherwise, you have from 30 days before to book stuff. It might help you relax a bit just to know that you have a few set plans, so you know when to try to plan a bathroom break. I know its easier than it sounds! But sometimes it's easier for me knowing I have a more set schedule.
The app also has a bathroom map on it! I think you can even have it find the nearest bathroom based on your location in the park, although I haven't tried it yet. If you don't use the app, pick up a map and take a few minutes to circle or mark the bathrooms on it. They're usually easy to find, and there are plenty of them around. Bring a change of clothes and some toiletries, too, if that would at least give you some peace of mind.
I've also gotten a medical pass a few times. I know they changed the system around recently, but if you go to Guest Relations in the front of the park just explain your condition and how it makes it hard to wait in line, and you might have to get off of the line suddenly. Then they give you a pass with the number of people in your family and what accommodations they can make for you. In the past, they would send you in another line for a shorter wait, or they'd let you back in if you had to run to the bathroom. I think now they assign you a designated return time (like a Fastpass), so you can go sit down and relax, then go back to the ride at the time. The medical passes kinda vary based on situation, so just ask and see how they can help. And definitely rent a wheelchair or something similar if it would help!
They do have a lot of plainer food options in the park these days, too. I try to stick to breads, pastas, yogurts, and sandwiches if I can. Drink a lot of water! Sometimes it's also nice to head to the park early, go back to the hotel midday to rest, and then head back into the park at night. Just take your time and don't be afraid to take things slowly, and I'm sure you'll have a great time!
Moss, thanks a bunch! I figured there had to be someone here who is a Disney regular! 
