First off, welcome to
Wild Texas!
I haven't been to Garner in awhile so from memory I'm afraid I can't recommend a specific shelter. There were many on the "fringes" that would be more secluded that in the middle of the mass of them. Also, generally as long as the park isn't absolutely teeming with visitors (and it shouldn't be right now), the ranger/staff will let you drive through the park after checking in at the ranger station, so you can scout a site/shelter and pick one, then return to the station to let them know which you've selected and pay your fees. We do this at most parks, and except for parks with cabins that's always been permitted to us by the park staff unless they're at peak capacity/visitation. Sometimes, we'll "cheat" and just ask the rangers, based on our criteria, which they recommend and you can't go wrong that way, either. Some of our best campsites have been ranger-recommended. They hear which sites people most liked time and time again, and are a good source of info. Not so much, the folks in Austin at the central reservation line, though if they're personally familiar with a park they might have a recommendation, I s'pose.