5 full days - and far more than 38 hours
Welcome to the RTA Forum!
I'm sorry, but you are seriously misguided on your plans. It's simply not possible to do what you've proposed, and you MUST take much more time to do this trip safely. I'm sure a major part of that is that you are believing some very bad information from online mapping programs that suggest you could complete this trip in 38 hours. In the real world, driving 12 hours a day for 3 straight days wouldn't even get you to Denver, much less to within 2 hours of Yellowstone - and even trying to drive that far in 3 days would be far more than we'd recommend.
Boca to Yellowstone is 2600 miles.. A professional driver would be required to take FIVE days to make this trip because of safety laws.
When looking at a trip of this distance, you need to think in terms of days, not hours. 550-600 miles is a very full day on the road (10-12 hours a day), especially for a solo traveler with little to no long distance driving experience. That also falls in line with the rules and guildelines that professionals with years of experience and training are required to follow. It also does not make sense to think in terms of "7 stops for gas." On a long distance trip, especially when traveling solo, it is very important to frequently stop and take even a short break from the road, just to move around and give yourself a mental break from focusing on operating a 2 ton machine at 70 mph.
I'm very excited for you that you'll get to spend your summer at one of my favorite places on the planet, but if you are going to drive there, you need to give yourself 5 full days to get there. If you can't take that time, then driving can not be an option for you, and you must look at flying.