Tex78,
The Texas Code of Criminal Procedure does not specify in what manner you may carry a gun while traveling. It merely states that it is a defense to prosecution that you were travelling while found illegally carrying a weapon. If you would like to carry a handgun I would strongly suggest you apply for a CHL, it is a relatively easy and painless process.
I would advise against the unlicensed carry of a handgun even while traveling to anyone. While you may not be prosecuted because you have a viable defense, that does not mean you will not be arrested in the process. Remember, police arent responsible for interpreting the laws, only enforcing them.
I say all this trying to help you or anyone else out, not trying to be a know it all. I do have a degree in criminal justice though, and my sister is an attorney so I know what Im talking about.
That being said, if I were to carry unlicensed when I were traveling I would keep it in the glove compartment. Depending upon the attitude of the officer, if I were pulled over, I might inform him of the gun or I might not. That would purely be a judgement call. While some officers would like that you were honest enough to tell them what was going on there are the others who might not be so impressed and would haul you in to let the judge "sort things out".
In any case, good luck and be glad you live in Texas where you can still own guns. I still get sick when I see what all the people from states like California have to put up with, but thats another thread.
Michael