Kerrville, Fredericksburg, and Llano have touristy downtowns and probably some event or fair. Leave early and take a scenic route through the Hill Country. Look into Pedernales Falls and Enchanted Rock state parks, but reserve a day pass at the exact time they go on sale, they'll sell out in minutes.

I drove to Beeville for the annular eclipse in October, lots of people in parking lots and on the side of the road, but no traffic of note coming or going. Not sure how that will extrapolate to a full eclipse.