I just did a cross country trip with my aunt. From Dallas, to Indiana, to Florida, and back to Dallas. It's an 01 Accord with 90 something thousand miles on it. She's never had any major repairs done to it, and just took it in to the shop for a check up a day before we left. I did almost all of the driving (besides from florida to texas again), and never had an issue with it.
I've driven round trip from Orlando to Miami twice in one week (that's about 800-900 miles) on a 01 Ranger with well over 100,000 miles. It now has 160,000 on it. And I'd drive the van that has 211,000 miles on it.
AAA is a life saver, and can be a major help. If you're that concerned, you can take your car to the Car Care Center outside of the TTC and have them do a once over on the car before you head back out on the road. If you take it early in the morning, it shouldn't take them any time at all. And they're always helpful in taking people to the TTC, so you can get on Disney transportation and come back in the afternoon.
You'll just wind up getting an oil change in about a month rather than 3 months later.