It appears based on the following website that a few months is reasonable to expect for the N-336 appear. Of course this article is from a few years ago when immigration was a bit easier:
N-336 Successful Naturalization Application
As for whether you should appeal or file a new case, I would still say you are better off filing a new case. I don't believe that one precludes the other so you can still file a new N-400. If an attorney does advise you that filing N-336 is the best course of action, of course I will defer to that attorney's judgment.
I'm sure you are aware of this, but "innocent until proven guilty", while true under certain circumstances, isn't universally true in all situations. Simply being arrested--even without immigration complications--has consequences like having to post bail, having restrictions placed on you, etc. However the way it is, in theory, supposed to work seems to be that whichever party seeks to change the status quo has the burden of proof.
So in the case of the criminal case, it was the prosecution that wanted to change the status quo, by punishing you for something you had not been proven guilty of as yet. So the prosecution had the burden of proof--and evidently that was a bar they weren't in a position to clear, as they dropped the case.
In the naturalization case, though, it is you who are trying to change the status quo (from green card holder to citizen). So you bear the burden of proof. USCIS is simply saying that you didn't meet that burden--not that you'll never qualify to naturalize. In the case I quoted above, USCIS seems to have made multiple very clear errors in denying the naturalization, and so an appeal by a competent attorney was necessary to keep them honest (or get them honest if they weren't honest to start with). I'm not sure that I see similar errors in your situation. The difference between the outcome of the two cases (the criminal case and the naturalization case) seems to have primarily been one of the burden of proof. In the criminal case, the government bore the burden of proof, but in the naturalization case, you bore the burden of proof.