First off, 8 patients in a step down unit is ludicrous even if none of them are vented. I don't see how one nurse is supposed to take care of that number of high acuity, task intensive patients in any 12.5 hour period.
Second, the whole disciplinary process in regards to the med errors was not handled very well by management. This is just my opinion, but for the purpose of disciplinary action, I think that the two med errors the OP was originally written up for should have been taken as one occurrence, hence, just a verbal warning rather than a verbal and a written. Also, given the ridiculous amount of time it took to do chart audits and bring these occurrences to light, I don't think the third med error warrented any additional disciplinary action. Unless there were med errors being made every week, it seems reasonable to institute a "grace period" from the time the first two errors occurred up until the time the first disciplinary action was taken.
Third, it is not at all necessary to include on a resignation letter that it is being offered at the request of management. That's going to remain a part of your personnel file and can be reviewed by any hospital you apply to in the future if allow them access to your previous records as part of the application process. Generally, less is better. If you are leaving under bad terms it is fine to just write something generic thanking them for the opportunity but that you are leaving to explore other avenues of employment.
Last, I'm glad you got out. That sounds like a floor that would just chew you up, spit your out and guarantee a fast burnout. Good luck at your interview!