If a Palo Duro type of experience is your goal, the NM state parks (see here
) are all very good. I've been to many of them and they usually have an activity attached - hiking trails or something. They use a very goofed up system (during the summer) which causes a reservation to be almost necessary if staying for more than one night.
Depending on the time of year, finding high altitude campgrounds may be important. I get as high as possible during the hot months as it makes a huge difference in comfort. The problem with this is that there is not much available between Amarillo (Palo Duro) and Albuquerque.
There are a bunch of forest service campgrounds which are usually pretty nice but dry camping. The Quaking Aspen campground south of Gallup (Fort Wingate - way west of where you said you are going) as wonderful in the heat of the summer as it is pretty high. Fantastic mountain bike and hiking trails all over the area. No hookups.
Since you are coming across from the east, it's probably worth hitting a spot or two in the Ruidosa/Capitan area - also high/cool. Though that is a bit out of your way from Palo Duro.
You have lots of options in the mountains around Santa Fe!