287 to 87 to I-25 is a very good route. We do it almost every year. But we typically stay at RV parks in Amarillo (Ft Amarillo RV Resort on the western edge of Amarillo) and/or the Raton KOA (nothing great, but not bad). There is also a small RV park at the very top of Raton Pass. Great view but nothing close by if you want to eat out or shop.
I know there is a rest area with large parking area just north of Hartley. And another smaller one just north of Texline. Both in pretty remote areas. But I don't know if they allow overnight parking or not.
We typically stay on I-40 west thru Amarillo to US 385 north at Vega to Hartley to hit 87. No towns of much consequence that route. Slightly longer than going north on US 287/87 to Dumas. But, I think, a little faster as you do not slow down thru Amarillo and you do not go thru Dumas. Although Dumas is not much, there are still traffic lights and slow speed limits. But really not much difference either way.
If you do go the Dumas route, take Loop 335 north at the eastern edge of Amarillo, loop around the northeast side, and hit 287/87 north of town. Otherwise, you have to go right thru downtown Amarillo, unless you just want to see it. We did once, and saw 3 saddled horses tied outside the Dairy Queen while their riders were have lunch.