Headlights won't turn off?

My battery has gone dead twice in a week after moving to a much colder state, I attributed this to the weather and an old battery at first until my boyfriend noticed that my headlights are staying on. Even when the doors are all shut and locked, keys out of the ignition, the headlights stay on. We pulled the DRL fuse and the lights went off, but then I can no longer use my headlights at all (park lights will come on). Is this possibly a relay or the DRL module? And if so, will we be able to change one of them ourselves? Neither of us are very auto minded, but we're also broke, much like everyone else. Thank you!