The OP asked for recommendations OTHER then Gruene and New Braunfels yet that is what people are recommending. The OP also wants places she can take her dog and every recommendation posted are places where dogs are forbidden. Some state parks allow dogs but never ever in the cabins. Their loss since 80% of Americans travel with pets.
But the poster who recommended The Grist Mill Restaurant I say thanks. I will put that on my short list of places to visit and explore. That place looks awesome !
Since you were suggesting areas of Gruene/New Braunfels then go here:
Little Rocky Lodge and Resort - The Jewel on Lake Whitney!
Dogs are welcome in the cabins and the rental boats and the lake is beautiful.
Most hotels today take pets. But most parks are still living in the dark ages and dont. If a place dont take pets then do not patronize them. They will get the hint some day. I would much rather be around all those dogs then screaming crying annoying bratty spoiled kids any day. Dogs do less damage then kids too.
Have a great trip.......