I run tire liners (Mr. Tuffy on my road bike and Slime liners on my MTB) and Slime (or something similar like True Goo). On my road bike I also have thorn resistant tubes, although I think the rear one isn't slimed. Since I've been running that combo I've had little trouble - only one flat in the last 1200 miles or so.
I should probably run something a bit lighter and put up with a few more flats, but I REALLY hate having to stop mid-ride to fix a flat so I put up with the weight.
My trailer is another story. I'm running slimed tubes with no liners and they're not thorn-resistant tubes (thorn-resistant tubes are made of thicker rubber, so it takes a longer spine to poke through them). I've had several flats on the trailer, so I'll probably at a minimum add a set of tire liners.
The Bosque trail is among the worst for goatheads, but you find them everywhere around here.
Be careful if you have to take your bike off the trail, even if it looks like bare dirt, and at a minimum I'd suggest putting in tire liners.
Armadillos are rather good, but I'd try liners and/or thorn-resistant tubes before you spring for a new set of tires. It's a lot cheaper, for one thing, and if you end up going for the Armadillos you'll probably still want the liners and thorn-resistant tubes. I've got a set of Bontrager something-or-others on the road bike and some inexpensive Forte knobbies on the MTB and have yet to have a problem. The Bontragers are about worn out, though, and I'll likely go for Armadillos to replace them.