It's a Cordia. Going by the flowers I think it's C. boissieri, the Texas wild olive: if it's regularly watered it will bloom pretty much non-stop, otherwise it will just bloom in late Spring-early Summer.
I t-h-i-n-k it would do well in your area, but be aware it will drop leaves in cold weather and anything below 15°F will kill it. Cordia are tropical plants and generally don't like cold weather, but C. boissieri is the most cold-tolerant of the lot, albeit one of the least showy species.
The chief problem is I've never seen one for sale: I saw a few in San Diego, so it's definetely raised in nurseries, but I cannot point towards any potnential source... maybe some member from Texas could help you.
