Maybe try casting to different spot and start your drift from a different area, even try casting from a different spot. Changing either one of these approaches may keep the eel up off the bottom just long enough to get your bait over the discarded line and hang-ups. It really sounds like you need to find a different area to fish eels
Another thing you might try is to use a foam bobber type of system that will let the eel swim naturally but keep it just up off the bottom. Good Luck