No, unfortunately to the best of our knowledge you cannot place a claim against the policy for the other car that the owner has insurance on for the damages he caused with the uninsured car. The driver though should still be held responsible for the damages he caused.
To get the damages repaired you, and the other damaged parties may want to place claims against your own insurance companies through your collision coverages. As for the carport, this may need to be claimed through your homeowner's policy, check with your agent in regards to making a claim for this.
Alternatively, you may want to seek legal counsel to find out if you will have to take your neighbor to court and seek a judgment against him for the damages you sustained when he was drunk driving.
For consumer help with your situation you may want to contact the Texas Department of Insurance online or call their consumer hotline at 1-800-252-3439.