When you have the red wire connected, and the headlights won't turn off, can you switch them between high and low beam? If not, it sounds like the relay points have fused together, and it needs to be replaced.

When the headlights are stuck on, do they go out when the key is turned off? They are supposed to, if the system is working correctly and you leave them turned on. This feature is both to save the battery if you are forgetful, and to make it convenient to use them as daytime running lights in gloomy Northern winters.

Is your stalk switch on the steering column working correctly? It is possible for this to jam in the pulled back (momentary flash of high beams) position. Sometimes it can be freed by pushing it forward. Then don't pull back quite so hard next time.