OP: You can use a cover letter, but the cover letter should not be too detailed on your career ambitions. A cover letter for a leadership position is not a scholarship
application. With that said, hiring managers want to know how hiring you vs. someone else will further his/her objectives for their unit. Therefore, if you choose to write a cover letter (and this is not a bad idea) it needs to focus on what you can or will offer the hiring manager.
What skills do you bring to the table? What leadership positions have you had in the past? Have you ever had direct reports? If so, how many? It is OK to use your burger flipping / retail experience if you have it... I have been passed over while attempting to be hired into my first management position by managers who preferred nurses that had direct reports as prior managers in other fields.
In addition, as a charge nurse, what have you done to improve your current unit? What have you done to make your facility survey-ready?
By the way, do you know of the initiatives that are being implemented on the unit to which you are applying? If not, it will help to find out prior to your interview.