The two at the motel inside the loop, I forget the name, are definitely was fully public last I saw.
The Holiday Inn Express right off 290 is a little more grey area but I've used it. You don't want to count on Level 2 (L2) charging unless you're traveling well off the grid, they are generally bail-outs.
Destination L2 are more for the end of your trip stuff, or meaningful waypoints. As the name suggests, "destinations". They are exactly like charging at home, using literally the same equipment if you have a Tesla HPWC in your garage. So they get added by places like hotels, motels, some large shopping areas, B&B places, wineries, restaurants, etc. There's a lot of empty miles on 290 itself.
Where do you go in Austin, what part of town? What time of day are you there? Until Giddings is done, the Austin SC, on I35 should get you charged in a similar amount of time to get you home. It definitely can be busy, depending on the time of day, but there's a new site NW under construction that should be open sometime this summer and hopefully will help with traffic at the SC on I35.
<edit> Oh, I'm sorry I had your trip direction backwards. You're going to do Austin-Houston-Austin. Yeah, there should be no trouble with making the trip on only a quick 15-20 min charge-up at the SC at Rudy's on 290, or Richmond Rudy's near 69 & 6, or the Tesla SC I45 Spring location, depending on where in Houston area your son is going, as you're leaving back toward Austin. That's an easy-peasy trip, it is the Austin SC that gives all us out-of-towners the willies coming into Austin.