Also any advice on a cross country move, or what I may forget to do before leaving, will be appreciated! I have ONE week to go home pack up a whole house and move 24 hour drive away!
I don't think it's possible to drive cross country in 24 hours, so I'm assuming you aren't going from one coast to the other. We did a cross country move from Maine to Arizona a few years ago and it took four and a half days, loooong days! This included stopping to sleep at a hotel each night.
My husband drove a large moving truck with a tow behind it for one of our cars. My daughter, cat and I followed in another car a few weeks later. Earlier, my daughter and I drove another car cross country, then flew back.
I agree with previous posters about the car tow. We used the kind that you drive the car onto and the entire car is off the ground. Also, check the car each day just to make sure all attachments are secure. My husband said that one morning when he checked the car, he noticed one of the chains (or whatever it was that held one of the front tires/wheels in place) had loosened.
A few more tips:
The trip will take longer than in a regular car because you just can't travel as fast.
If you're traveling through an area that has toll roads, it will cost you more (sometimes quite a bit more!) because you pay by the axle.
Utilize the rest stops that are along the interstates! My husband said it was much easier pulling off and back on the highways, and parking, in the rest areas, rather than pulling off an exit. Also know that many rest areas have been closed due to the economy (can't afford to keep them staffed). Some of them were temporarily closed and have reopened, but many remain closed. So, if you see one open and think maybe you can "hold it"
or go further before getting gas, you might want to reconsider!
One final tip: If you are driving through an area that has border patrol stations, and agents directing you to pull over, make sure you do. They aren't just waving at you to be friendly! My husband learned this the hard way.
Being from New England, we weren't accustomed to going through border patrol stations. It was late at night and my husband evidently didn't see the signs (or the frantic agents waving at him) and at the last second realized they were shouting at HIM to stop. The agents had my husband get out of the truck and they did a complete inspection, even using dogs to sniff in and around the truck and car. I guess they thought he was trying to avoid stopping. He told me afterwards that the agents were none to pleased and one even made a snide remark about people from Maine not being able to read signs.
So, that's our driving across country in a moving truck story!
ETA: The following year we returned to Maine for most of the summer and drove cross country again. The second trip we towed a small U-Haul filled with stuff we had left behind. That was an entirely different experience! We had to drive below the speed limit most of the time because if you went too fast, the U-Haul would shift from side-to-side. I found it extremely nerve-wracking and I was constantly watching the trailer from the rearview and side mirrors, even when I wasn't the one driving.
. Don't want to cozy up to the wrong pump and realize you have to move and try again. I would also stick to the outer aisle pumps to mitigate the risk of hitting one.
We had to figure out for ourselves how to get the straps tightened and untightened. Somewhere in the neighborhood of 30-45 mins. it took us
And the wires connecting the signals were not the best either. My dad is a retired electrician and he shook his head when he saw the wires. 
