Do you have a job offer? What kind of application are you looking to do?
Were you legal in the US for 1 year even if you are not legal anymore or are you a tourist there?
You could go to Canada as a tourist, look for a job, wait for the LMO and then leave the country just to the US to get your work permit issued but it could take up to a few months to get the LMO and work permit. It might not be a good idea to wait for it here when you are spending money and not allowed to work.