You don't have to stay on the interstate the whole way.
Try state # 9 in New Mexico, Straight shot across , with little traffic.
62/160 in Tx. takes you the Guadalupe mt.s and it's a short hop to the Caverns from there.
Lots of roads in Tx. are pretty straight with fairly high speed limits.It's kind of off the path , but, have you considered Big Bend ?
RR's Catnip Hill to Peoria ___Loopin' Seattle to WestFest
It started with some beers __1500 miles to the Dentist
Skeedaddle to Seattle______ A 30 year old on a Three Flags Run