I pleaded to a continuance without a finding to my DUI charge. It is the only thing I have ever gotten in trouble with the law for. It is not a guilty charge, but CWOF shows up on my record. Will I be able to travel to Australia?

Are you still on probation? If so, no you can't travel to Australia. In any event, I don't think you can travel with a CWOF because you have filed for a continuance, but if you go to court again, you can still be found guilty.