Source: "Only a few hours after those on the coast began fleeing the area news outlets were reporting that given the sheer numbers of evacuees there was not a hotel room available in the entire state of Texas. This means that people without relatives or friends outside the storm zone would have to go all the way out of state to the north or to the west to escape. The staggering amount of cars trying to flee the area means that such a trip could easily take 24 hours to get to a safe destination. Right now every single route out of the area is bumper to bumper in traffic that is barely moving, trips through Houston of just a few miles are reported to take over 5 hours! For many in the area there is simply nowhere to go."
I am fairly sure that if a large number evacuees tried to get into Mexico that they would quickly overwhelm the border infrastructure, though it does seem to be were people are NOT going. It is kind of like if you want lunch at noon. Where do you go? The worst place in town, their is always room there.