If a bad headlight relay is a reason your headlights won't turn off, the fix is to replace the relay. This is a little easier to check since there's a chance that multiple circuits may use the exact same type of relay. From your description it certainly sounds like your headlights are out of aim/alignment. Check power and ground connections for high beams that act as low beams, high resistance on either can cause that condition. The relay is the second from the left in the fuse box.