I wouldn't recommend crossing the Texas/Louisiana border anywhere but where the southern tier route shows you should. I crossed near Orange, TX and got clipped by a pick-up truck mirror in the county the cop told me has one of the highest DUI rates in the nation. The crossing further south on the Gulf Coast involves a long, very dangerous bridge that another cyclist told me should be "avoided at all costs".
I just looked up where Mescalero is and might have some advice on getting there. From Van Horn, TX, I departed from the southern tier route and headed straight north to Guadalupe Mountains National Park, then Carlsbad Caverns and Carlsbad, NM, then turned west to head up--and I mean up on a long, steady climb usually into the wind--to Cloudcroft, NM. The descent from Cloudcroft to Alamogordo is wicked fun and includes I think the only road tunnel in New Mexico. The contrast could not be greater between national-forest-surrounded Cloudcroft and desert-surrounded Alamogordo is stark. From Alamogordo you could probably head up 70 to Tularosa then over to Mescalero. Looks like there is another way to get there via S. Tularosa Canyon Road, but I have no idea what that road is like.
If you're into camping it might be worth sticking around Guadalupe Mountains park for a bit. It's what's called a "backcountry" park, as there aren't roads into it. There's a nice campground by the visitor's center--which has wifi!--and the rest of the park is trails. Set up a base camp there and do some hiking. Carlsbad Caverns is also amazing but requires a long climb up to the visitor's center. Once you are in Cloudcroft you can head south on some hilly, mountain crest riding to the National Solar Observatory. Not a must-see but beautiful views out across the basin and a cool place if you're into astronomy.
Alamogordo has a museum dedicated to space exploration which was quite fun. The grounds around the building is littered with rockets and capsules and other fun things. White Sands National Monument is worth a stop, but in both Alamogordo and White Sands beware dust storms. They can appear quickly and reduce visibility to near zero. Dust gets everywhere, too, so consider putting electronics in ziploc bags and not taking them out until you're out of the area.
You can see more on maps and things on the crazyguyonabike link above, under the "New Mexico" section.
And I second the advice to do the Gila Hot Springs loop. There's a campground with hot springs water for soaking.