May just be your pit. Those pits are lighter gauge steel and just don't hold the heat. I start my fire with lump mesquite that I purchase at Sams in 40# bags. Usually pretty large pieces. Let the fire burn for about 45 minutes to an hour to get the pit to temp. During that time I will throw on a couple of splits, usually mesquite because that is what I like as well.
After that it's a matter of regulating the draft to get the temp I want. Usually 300 - 325. I then add lump as needed. You can't wait until the fire gets too low to do that. I will also throw on another split as needed, say every hour. I will do this until I foil at which point I just use the lump or throw on other wood such as oak, which I don't like as well. I do not find that I get excess smoke, but my splits are well seasoned and stored in a dry place.
On another note you are in the heart of mesquite country in Ft. Worth. It shouldn't be too hard to find mesquite for purchase. Once it is dry you can split it easily with an axe or maul. Or if you know someone with a hunting lease you might see if they would cut some for you. I bring back a trailer load a couple of times a year from the hunting lease. The ranchers hate mesquite and usually don't care if you cut them.