There are no cabins at
McKinney Falls, but there are screened shelters.
Inks Lake State Park is larger, has more accommodations and access to water, but also has no cabins -- they do have new screened shelters that are nicer than some of the other parks' shelters -- definitely a couple steps above just a shelter with a bare concrete floor. Also, there are some privately owned/run guest houses and cabins in the Inks Lake vicinity. I think I have some links for some in the area, though some were damaged in floods and didn't resume operations back in '03 or so.
Have you checked into
Bastrop State Park? Justin and I have stayed at the cabins there on several occasions and they're nice, though at short notice you are probably unlikely to secure one. They have kitchen and bathroom facilities, some have fireplaces and multiple bedrooms. It's been several years since we've stayed in one, however.
Also, consider
Garner State Park. Again, no cabins, but plenty of nice screened shelters with bed linens and the like provided, and a nice cold river to swim in with beautiful scenery. Very kid and family friendly park (very busy in summer and Spring Break).
I have been out of the loop lately (no camping/hiking), so maybe some of the regulars here can guide you more in the right direction. Sorry I can't be more helpful. I've got a lot of catching up to do, as evidenced by the utter lack of trip reports from me in the past year or so, before my knee surgery.