Boy OK people throwing you under the bus over here. First, resigning really won't help you because they can report you to the BON anyway. Trust me on that. I am a nurse in recovery, caught diverting controlled substances and had to report my own self to the BON. I had a hearing 3 years ago with the BON and still have an active and clean license because I was graciously offered an alternative to discipline recovery program. Anyway, I go to recovery meetings with fellow nurses in recovery and have heard it all as far as being fired and reported to the BON (for drugs/alcohol).
Quitting does not save you from being reported whatsoever.
It is very possible they will just fire you and not report you, however.
They might not even fire you.
I can't say this is 100% always the case, but at least in my area hospitals literally only confirm dates of employment and title held during reference checks. I know of many nurses in my recovery group who were fired and even arrested for diverting and not one of them had a problem putting their old employer on their resume/references because like I said, in my area at least the hospitals HR policy only allows dates of employment and title held to be given out for a reference check. The exception to this is only if it is another affiliated hospital... then they will say if you are "rehireable" or not... and quitting vs being fired won't change the "rehireable" status. So basically if you are fired you would have to look for work outside of whatever system you currently work for.
Getting fired has not negatively impacted any of the nurses I know from my group getting a job, but getting in trouble with the BON has if they decide on any form of public discipline.
If they do report you, you will likely have a long time before the actual hearing and nothing will happen to your license until then. You could wait even a year or longer because the BONs from most states are backed up. In the meantime, your license is clean and you can get another job while waiting a BON hearing (worst case scenario because they might not report you).
If it were me, I would stick it out to see if they fire me or not, as I see no benefit to quitting vs being fired. That is what I personally would do. But I do recommend calling lawyers who specialize in nurse licenses in your state for further advise.