What is occurring to your watermelons is called "blossom-end rot." Even though you say that you are watering frequently, they must be drying out too much between waterings. This is what Dr. Sam Cotner says in his book
The Vegetable Book:
"Blossom-end rot of watermelons is very similar to the same problem that occurs on tomatoes and primarily is related to a soil-moisture imbalance. Mulching and occasional watering to prevent moisture stress during dry periods will help prevent blossom-end rot. If the problem occurs, prune off all the affected fruits."
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