stan,
I have had cars shipped several times to Florida from different parts of the country but never to Texas. However, of all the companies that I have used, far and away the very best was Exotic Car transport Specialists. All are enclosed if you ask for it, and each car rides on it's own shocks. They are very good about inspecting the car body etc for damage before hand and note each very well. The down side is that they are more expensive but I do feel that for a car in good to great shape, that it is well worth it. I have always been very happy with their service, and have been burnt by some others.
Their web site is
www.exoticcartransport.comThere is also a website that reviews autotransport companies which you should google for which I highly recommend but can not recall what it's name is at this time. It has posts from people all over the country about their experience with different auto transport companies.
Do be aware of a company from Washington state, I don't recall the name right now but they are listed under several names and have a awful record and my experience is included under one of their company names.
Good luck.