We've run coast to coast 4 times so far (snowbirding in Az and Ca) and can only tell you our experiences in January (going) and late March/early April returning:
I-40: gatta watch the weather...and few cg's are open in the winter. Really gotta search. Winter storms drive RVs down to I-20.
I-20: Better than I-40 as far as open cgs...but the road itself seems always to need major repairs. And here, too...winter storms drive people further south to I-10.
I-10: NEVER a problem finding open cgs'. The southernmost route coast to coast. I suggest however, you figure where you are going to stay the next night and call/make a reservation. We travel about 300 miles a day. Using the KOA book, I figure we run from KOA to KOA about 80% of the time. ALL easy-on/easy-off. If a KOA is not around, we use the Trailer Life Directory...altho I question some of their ratings....we were just fine.
With I-10, if you're not going to New Orleans, I suggest you take I-12 around it. Much better road.
Enjoy your trip!