Personally I'd rather avoid the I-10 from El Paso to Fort Stockton drive.
So we usually cross that part of the country on US-70 from Las Cruces and Roswell into Texas.
But we do at times like last September. Now, I cheat. Being retired Navy, I stopped at the Fort Bliss RV Park. And left I-10 at Van Horn for the much friendlier drive on US-90.
In El Paso, the TX Franklin Mountains SP is not someplace you want to try to take an RV, and while Hueco Tanks SP east of El Paso is interesting, probably not a good base to explore El Paso.
One the NM side of town, I'd recommend Las Cruces. There is a park that looks decent about halfway between Las Cruces and El Paso but along that stretch of I-10 are a bunch of dairies. If the wind and weather is not exactly right - the smell and the irritating things in the air will clean out your sinuses, and maybe take the paint off your rig.
I like NM state parks. Wish they had not forced me to pay a bit over a hundred in cancellation fees last summer and fall when the state parks were closed, and then only opened to NM state residents. They cancelled 12 reservations, and while the camping fees were refunded, the reservation fees were not.
NM reopened state parks to out-of-state visitors on Feb 27 for camping.
Alamogordo is too far to commute back and forth in my opinion, with Las Cruces as an alternative.