I rode from South Carolina to Abilene, TX, wild camping the entire way without a single hotel stay. You won't have too many problems, just scout out a campsite about an hour or two before sunset, you don't want to wait until dark. You should also learn how to do basic repairs and maintenance on your bike and carry extra tubes etc so the answer is yes, take the proper tools, don't rely on an LBS being nearby. The catcus (cacti?) in Texas seem to shoot needles in your tires from the side, I suggest slime or really flat resistant tubes/tires.

As far as the weather, yes it can get cold in the ST, and it can get bitter cold in western Texas and New Mexico in December/January
An example is Van Horn, Texas
http://weather-warehouse.com/Weather..._December.html
Look at those minimum low temps and average low temps for December... BURRRRRRR