I don't know any begonias that want full sun, especially down here where the sun is so potent. Maybe in the beautiful PNW where it never gets very hot and the sun is usually hiding behind the clouds. (We say that, those of us who love the PNW, to keep its beauty to ourselves, even when we have to leave it.) I don't think your sun in San Antonio is a bit gentler than ours in south Louisiana. Some canes will take quite a bit of sun, but not full.

Semperflorens will take full sun. These are what I call "landscape" begonias. They aren't the fancy rexes or rhizomatous types if that is what you were thinking about.