Schools do give out scholarships -- they range from full tuition to partial tuition to smaller awards. I was offered a $2500 scholarship to Baylor and a $5000 scholarship to UTHSCSA -- they were both based on my undergraduate performance.
Houston also gives out scholarships, but I am not sure in what amounts -- they do not award scholarships until school officially starts because they don't want the scholarship to be the only reason you choose to go there.