John-
Must be a typo error on your request for highway routes. US77 exits I-37 at Calallen on the west side of Corpus Christi to the RGV. US281 would be about 15 miles shorter. Although not totally interstate type, both highways are good roads and mostly four lane all the way to the RGV. My suggestion would be US281.
As far as campgrounds, there are some scattered along the way, recommend that you check something such as Good Sam's CG Directory to find a place that you might want to overnight if the second leg of your trip becomes too much. There are not many places south of George West on US281, better selection would be on US77 in the Corpus Christi or Kingsville area.
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