We did a very similar trip in Sept last year.
Space centre was fascinating (do the full tour, expensive but worth it) however Houston itself didn’t really do it for me. Maybe fly into Houston, do the space centre then head out.
San Antonio was smaller than I thought it would be but the Mission & the history was interesting. They’ve also redeveloped the river front there which is beautiful but, man, it was hot!
From there we did a few days in hill country for wineries & a bit of country chill out - the German heritage around there was surprisingl.
Austin was awesome. Cool city, v historic downtown, great burgers & beer.
Dallas Fort Worth was actually a ton more interesting than I thought it would be, JFK book repository museum, Baseball, Dallas Cowboys stadium tour, Southfork, cattle drive in FW etc.
We hired a car, self drive and flew out of DFW at the end. Not sure how easy it would be to do on public transport.
HTH