Could be woodlice, especially if you are mulching them with woodchip. Like you I have to pick them early because they get to them first. They ripen faster in a bag with some bananas as they give off a lot of ethylene gas as they age.

That rot should be helped with more calcium. Just leave some egg shells in vinegar for a few weeks dilute it 1 part per 1000 and spray it on..... As that takes a few weeks to prepare perhaps trying adding a couple of tablespoons of natural yogurt/ kefir /sauerkraut juice to your watering can next time you water. Lactobacillus usually counteracts a lot of rot issues.