If you weren't convicted via Court Martial, you were most likely administratively discharged under a General Discharge. I think that your chance of getting your General Discharge upgrade to a General Discharge Under Honorable Condition is slim, but if you feel that the investment is worth the cost in legal fees, I'd go for it.
The senior officer involved in your discharge would have had you sign documents acknowledging that accepting a General Discharge could cause you problems in your post-military life. You should have been given a notarized copy of those documents as part of your discharge package. If you didn't sign those documents, or those documents aren't in your military records, your position might be stronger, but you won't know unless you look.
