Given that you are clearly in violation of one of the department's stated drug policies AND you have virtually identified yourself (i.e. you have given your age, graduation date, identified the THREE departments you will be applying to and divulged your drug history in detail), your best bet at this point is to wait the three years.  Keep in mind that the information available on this site, although subject to abuse, is not meant to help people "bypass" policies for employment suitability.  Rather, the information is meant to be used by truly qualified candidates that do not wish to be accused of lying only because of a flawed polygraph "test."

Although your concern is understandable, your question is quite cut-and-dry.  You are not qualified for one of the departments you would like to work for.  I'm going to venture to say that just about any department/agency will DQ you for drug usage within the last year, regardless of which drug or how many times.  You SHOULD NOT apply.  Look at it this way - either wait three years and most likely be successful or apply now and seriously jeopardize your potential for a law enforcement career.  The polygraph is not your only hurdle.  Say you did pass a polygraph examination.  What happens when a background investigation uncovers your drug history (which you most likely lied about)?  A wall is then dropped between you and law enforcement.  Even if you are thinking, right now, that the only two people in the world that know about your drug usage are yourself and your girlfriend, what happens if your relationship with her goes sour and she decides to seek revenge?  A quick phone call to your employing department, although not necessarily a definite termination, will raise a lot of doubt and cause your credibility to be questioned.  Not good.

I don't know the specifics of your relationship, but I say lose the girlfriend, disassociate yourself with anyone that uses drugs to your knowledge and do something worthwhile and constructive for the next three years that does not involve illegal substances.  Take that path and you will likely meet success.  Although you may regret your drug usage and have no intention of ever using again, you are the only one that knows that right now.  Go three or four years without any more usage or association with drugs and others will begin to see this.  Good luck.