From Abiliene, take I-20 west to Monahans and pick up Hwy 18 to Fort Stockton then 385 into Big Bend Nat'l Park. Rio Grande Village campground is very nice.
Leave Big Bend either via 385 or 118 to Hwy 90. Take Hwy 90 to Van Horn, TX then 54 north to 62/180 east to Carlsbad, NM.
To get to Alamogordo, NM you'll take 285 north to Roswell then 70 to Alamogordo/White Sands Nat'l Monument. Definitely don't take Hwy 82 from Artesia to Alamogordo - very steep.
Another way to White Sands from Carlsbad would be to backtrack on 62/180 to El Paso and Hwy 54 north to Alamogordo.
From Alamogordo to Phoenix in winter it's best to take 70 to I-10 west.
You won't have any big hills on these routes but will have ups and downs. It's not a completely flat drive.
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel