I have driven all over the USA many times in a big rig class A with a 4 down in tow and the little grades between Chicago and FL are or no concern for any traveler of the Interstate highways. A 30' RV will not be much different than driving a car.
If a refrigerator is needed to store medicine the one item I would recommend is to make sure that there is Propane in the tanks before beginning the trip. I bought a new rig once and the dealer assured me that the tank was filled and it was not. Humans are imperfect and do make mistakes.
I recommend highways 65/24/75. Easy Internet drive and allows plenty of short side trips to meet your schedule. Just take the BELTLINES AROUND the major cities.
