Get the nose clips. Eventually you won't need them any more, but in the meantime they'll help you get through the race.
Can you swim breaststroke? Or, should I say, can you swim 300m breaststroke in under 10 minutes? The breathing pattern in breaststroke is pretty hard to get wrong.
If breaststroke isn't going to work for you, one thing you might try is practicing your breathing with a kickboard. Kick with your face in the water, and just concentrate on one arm at a time, turning your head to the side as your hand reaches your hip. Once you get comfortable doing this lose the kickboard and see if breathing during your normal stroke is better.
By the way, I assume when you breathe you're turning your head to the side of the arm that's at the end of the stroke, right? By this I mean when breathing to the left, your left hand should be at your left hip and coming back up.