Inadequate pollination probably.

I've had to hand pollinate all my cucurbits, otherwise they just shrivel up or rot while still very small.

Normally, the flower rotting is normal--it serves its purpose to pollinate the fruit and then rots and falls off.
If the fruit is rotting when the squash is still small, then as Peter says, it's because the fruit didn't get pollinated so it is dying.
What you have to do is, in the morning when the flower opens on the female (the fruit) pollinate it with pollen from an open male flower.

If you're getting rot when the squash is bigger, then I don't know what is going on.

It might be blossom end rot. How big is the squash when it starts molding/rotting?

Rodney