Problem with reference from previous employer

Hello, this is my first post here, hope everyone is doing well. I was wondering if anyone else has had this problem or how they handled this type of situation. Six years ago I worked at a hospital and was caught diverting a prescription. Bascially, I resigned in lieu of termination. They pretty much gave me no choice by saying I could stay employed there and transfer to a different area but they would turn me over to law enforcement and take legal action for their own protection unless I resigned. I of course resigned to avoid legal charges. They did report me to the BON which I understand is mandatory. Before the board even received the complaint I enrolled myself in the NAP monitoring program. So when I went in front of the board, I was already enrolled in the non-disciplinary track of the NAP program, and the board decided I could remain in it for 3 years with no official disciplinary action taken against my license as long as I successfully completed the program, which I did with not a single positive UA, I actively participated in my outpatient counseling and have been clean for 6 years now :) I was fortunate enough to get hired at a LTCF while in the NAP program and have been there for 5 years with glowing reviews but it is a very difficult, stressful job, the working conditions are horrendous for the most part and I am ready to move on and find something new where I can see myself retiring from. The problem I am running into is I have had two job interviews in the past 3 months, both of which I was pretty much told the job was all but mine until it came time for the reference checks. I suspect the hospital I diverted from is disclosing this information. ON my applications I always say I left due to a medical illness that is now resolved (which is true..addiction is an illness and I am in recovery). There is absolutely nothing on my license, no actions, it is perfectly clean, the BON told me I never have to disclose the incident to future employers, it will never be brought up again unless I get another complaint in the future. When I left the hospital I met with the VP and the manager or HR and they assured me this information would never be disclosed during a reference but I can't help but wonder because as soon as I get to that point and they check my references, I am not being offered these jobs. Unfortunately I never got anything in writing, this was just a verbal understanding. Any advice on how to handle this would be appreciated. I have been thinking of contacting HR at the hospital. I have been thinking about just leaving the hospital off my resume/applications but I don't know if thats a good idea (what if they found out I worked there and then it would look like I'm dishonest/hiding something, plus I want employers to see that I have that experience. Something is holding me back here and I am getting very discouraged.