Can I get a state jail felony charge dropped to a misdemeanor charge?

Possession of a dangerous drug is a Class A misdemeanor. But, if the drug involved was cocaine, I assume you were actually charged with possession of a controlled substance. The prosecutor CAN almost always reduce a felony level charge to a misdemeanor. The real trick is convincing him or her to do so - they aren't likely to do it just for the asking.

Why has it taken so long for you to take a plea? If you have not already plead and the prosecution is not offering to drop it by now, I seriously doubt that if they have a good case against you they will drop it to a misdemeanor. If it is a very weak case, and not knowing the facts it is hard for me to say, they should just dismiss the case. I would also think that your area has a special drug court. If you qualify, most of the case end up with you case being cleared upon successful completion of the program. Best of luck.

Since you got the other two answers which discuss a pending charge, I'm going to answer this as if you already got probation (or prison) for the state jail case. I frequently have people call and ask if they can get an old case reduced. The short answer is no. In order to "go around" the disposition that previously occurred in the case, you would have to file an application for writ of habeas corpus (a complicated matter) and have it granted, then be brought back to court on the charge and work it out for the reduced charge. (Hope this helps.)