I think it depends on your age, your attention span, and your bladder capacity. Your biggest problem is going to be if you are driving alone.
A couple of years ago a friend of mine and I drove to western Iowa. We drove straight through, stopping for food, gas, bathroom only. We split the driving into 2-4 hour shifts depending on how we felt. It took about 24 hours out and 21 hours back.
I've also driven from BWI to southern AL, splitting the driving with my son. 21 hours straight through.
Boston to Austin with my daughter was a three day trip, but part of that was because of an issue with her truck and part was because of an issue with her cell phone. I could probably shave 6-8 hours off that trip now. In that case, I did all the driving, 12-14 hours the first two days, 8 the last.
I happen to enjoy driving, so it's not a big deal for me. If you don't enjoy it, it's torture.
I'd advise against having your car shipped by truck. Depending on exactly where you are going, it could take up to three weeks for your car to actually be in your hands. The car will go to the trucking company's depot, will sit there until they have a full truck to your area, and then will be delivered to you. That happened to my daughter in law when they moved from NJ to AL. They let the moving company handle the car move, they subbed it out to some car transport company, and it was three weeks before she had her car. In the end, the trucking company paid for her rental, but it took a lot of phone calls and yelling to get that done.