Best RV route Palm Springs, CA to Denison Texas

I will simply comment on the Texas end of the route.

An alternative would be to take US 281 north off of I-20
then @ Jacksboro take TX 59 all the way to St. Jo where you would
join US 82 E to Sherman where you could take either US 75 N
or TX 91 on in to Denison.
This will avoid the traffic congestion around DFW and bring you straight into Denison.

There are other ways to get up from I-20 to US 82, but this looks like the best.