I also live in Texas and have had the same problem with black spots on my watermelons, in many cases MUCH worse than your picture indicates.
I have been trying to determine the cause for 4 or 5 years now.
I have been told "over watering", "Under watering", "The soil is to rich" "Not rich enough" I think I have heard it all.
I even tried bedding the melons on straw and plastic plates.
This year I planted my melon beds in the same place I did last year.
Note: Last years melon crop over 90% of my watermelons had these black spots.
This year I have ONE watermelon so far with a small black spot, I have changed NOTHING in my planting and watering methods except, I planted Cantaloupe melons closer to the watermelon beds this year than last.
I have learned after last years planting that most of the melons will continue to grow with the bad spots and that the bad spot can be cut out at harvest and the melon was still sweet ripe and tasty. However you may lose a good portion of the melon in some cases.
I also tried removing the bad melons as soon as I detected a black spot and yet every melon to produce after that STILL had the black spots.
You have nothing to loose by trying to harvest your melon, Watch to see how it grows, if it dry's up and dies you have your answer, it may continue to grow and you can harvest a good portion of the melon.
I too would love to hear what others have to say about this problem.
Best of luck to you, I hope you watermelon turns out sweet and tasty.