Here is my take on your question about camping in Yellowstone N.P.
You are towing all the way from Indiana to visit one of most outstanding National Parks in the Country.
If you cancel your reservation and try to go Pot Luck on finding a camp site during July, a very busy month for the Park, you may come up empty.
Then you could find that the private camp grounds in West Yellowstone are booked. Camp grounds in Cody and the Tetons are long drive each day if you luck out and find one.
What would be left for you to try for a the few dry camp grounds that are around the out side edge of the park. This also would mean a long drive in and out of the N.P. which is huge. There are 360 miles of paved roads and trails in the Park just to give you an idea.
I would suggest that you keep your reservations and enjoy your time in the park, not driving back and forth.
If that Trailer has not moved for 11 years, I would not move it an inch until I checked the wheel bearings. If they are good, they will still need new grease packed into them.
200 miles on a dry bearing could cause it to freeze up on you and that will ruin your day.
Also, inspect the brake shoes to make sure that they will move and have not rusted in place.