If the screws have rotted away, try moving the new lights around side to side top to bottom, and see if you can cover to old screws and have good wood to hold the new light screws!!
Another option would be to dig out the old screws, purchase a can of talcum powder, fiberglass resin and hardener. Combine the thre to make a filler, fill the holes and apply a thin smooth coat, let it harden and the you can redrill the holes and attach the new lights! This definitely will work to fill the rotted area af wood!