My sympathies for your difficulties with your carrots. We’ve lost a few here at the kitchen garden to birds and mice, but it sounds like you have a big slug problem. We were warned that this would be a bad year for slugs following last year’s very wet weather, and you would appear to be paying the price.
I would recommend a dose of nemaslug on to your soil to kill off the population of slugs that is obviously living comfortably in your soil. Following that I would grow your carrots within a barrier such as slug barrier to stop any further slugs moving in. Another alternative would be to grow your carrots in a deep container instead of the ground where hopefully your slug pellets will cope better.
Good luck. Here’s hoping for some carrot success!