I spent a season working in a packing house for watermelons. They’d come in by the crateload and we were allowed to just grab one to eat any time we wanted.

The trick I was taught, and which proved to be pretty reliable over the course of the season, was to feel the veins. (This is possibly what’s being described as webbing here?) Watermelons aren’t smooth, they have wide “veins” running top to bottom and you can feel them if you put your hand flat on the side of the melon. The bigger/poofier/wider the veins, the more ripe is it.