First, the leaking is not caused by overproduction. The letdown mechanism is just a little too sensitive in some women at first. Second, the leaking will settle down over time, as the mechanism fine tunes and everything sort of "stabilizes."
Next, take a close look at the nursing pads. They should not be plastic backed. Cloth pads that you can wash and dry with the diapers are the best. Change them frequently during the day so the skin is not sitting against damp cloth. Try to have an hour a day of "air drying". These changes will make the skin much happier and may make it feasible to wear the pads at night, which is the easiest way to protect the bed.
If it's still impossible to wear the pads at night, sleep on a towel, possibly one folded a few times for more thickness and absorbency. Wear a soft Tshirt to help keep the milk somewhat confined. (Sorry to freak out the uninitiated, but milk can occasionally spray out and that would mean a larger part of the bed gets wet.) If the volume is dramatic, look for a large bed pad (designed for bedwetters) that is plushy above and waterproof below. These are washable also.
Be patient. The leakage will settle down and the skin will get happier too.