Have Military Law Questions? Ask a Military Lawyer.
Upgrading an OTH is extremely difficult because the standard for getting an upgrade is very high.
You have to file a DD Form 149 with the military records corrections board, alleging that the discharge in question was either given in error (for example, all your paperwork said you were going to get a General, but they accidentally gave you an OTH) or you have to show that it was an injustice that you got the OTH.
Now, if you had evidence that refuted the charges against you, that could be used to establish that it was an injustice to give you that discharge status. The problem though is that your facts make it seem like you signed an agreement to accept an OTH in lieu of having to go through a court martial. The board is going to see that as problematic, because you agreed to the discharge status rather than taking your opportunity to prove your innocence in court. At least, that is how they will likely take that fact.
Regardless, this is the only method by which one can upgrade an OTH if the discharge is older than 15 years (if it is less than 15 years ago, you could file a DD Form 293 with the discharge upgrade board, but they apply the same standard of review).