Canyon Lake area is nice, I live about 10 minutes away. Check on VRBO.com (vacation rentals by owner) during the winter cost should be reasonable. Check also on River Road, ( on the Guadalupe River betwenn Canyon Lake and New Braunfels). There are RV Parks and you could rent an RV here and set up on the river or the lake. It's a beautiful area and it's nice to have winters you guage by how many hrs you had below freezing. I appreciate it a lot because I work in North Dakota and their brutal winters.
I have stayed at the River Rim Resort in Concan. The wife and I both enjoyed it. Not far from the three twisted sisters.
DM what part of Laporte? I live in Fairmont Park close to Underwood.
For the op. You can find plenty of places to stay , but the cheapest route would be toy hauler. Google hill country rentals I found thousands of places from furnished long terms to weekend specials.