There are a number of one-day 6-on-6 tournaments; I think there's one in the Houston area, we've had one here in Seguin (used to be the day after our clinic, but I think we're going to move that to another date in 2012), and a number of them in West Texas.
Some squads have competed in the 7-on-7 format; will depend a lot on where you are. We did one at New Braunfels Canyon several years ago and ended up playing lots of 5A/4A schools. I think if you can get into one with 1A/2A level schools you might be at a better competition level.
Seems I heard one of our public brethern played in 7-on-7 last year and made it to the State tournament in College Station last year ...
You'll see info on the 6-on-6 on these message boards in the spring (most are in May and June).
As for the 7-on-7 qualifying tournaments and leagues, I think there's a website that handles them. I also think some of the crowded field private schools in Houston and Dallas do their own independent 7-on-7 tournaments/leagues. Most go from the end of school to the end of June or first week in July. I think the state tournament in College Station is in mid-July.