Audacity is known to have some built-in sync issues. Not latency necessarily, but the track is written just a hair later than what you recorded, so I always have to go back and sync it up after recording over a track. Not bad, just part of the game.

However, a 1 second delay would have to be something else. I guess I'd try to avoid monitoring yourself while singing if you don't have the equipment. Just record the track without putting yourself back through the speakers during recording.