The Lawson hammocks are heavy and cumbersome. I would return the hammock and get a better one. There are plenty that you can read about on this forum that you will like much better. ie. WBBB, Hennessey (I love my HHDJ), Wilderness Logic, and many more. PapaSmurf will custom make you one, and they are sweet.
Yes, an seperate Tarp will give you much better coverage. In Texas, you will need coverage from the rain and the sun...
Good luck, and welcome aloft from SC.
Also, you could make a hammock pretty inexpensively, and make them for the entire family.