Red Rock is right. This is a grounding problem, and one or both of your headlights is no longer grounded correctly.
Turn on the headlights and look at each bulb to see which one is burning both filaments regardless of the position of the dimmer switch. That's the one with the problem.
There are three wires in the headlight connector. High beam, low beam and ground. The ground will be short and bolted to the body somewhere near the headlight. Check that wire and its connections on both ends.
