You're looking for reassurance, and that's natural. It's also a sensible measure:  who can afford to spend time, money, and effort in vain?

  I know some members here will say, "perhaps a lawyer can expunge the conviction from your record,"  and that may even prove to be the most practical solution for you. But one thing that bothers me about expunging a conviction is the very question you mention in your posting:  "Have you ever been convicted of a felony?" How will a future employer choose to interpret a "No" response if your past becomes known? H. Ross Perot is the only employer I can recall who stuck his neck out for an employee.

   Here's something else that bears looking into--a mandated background check as a requirement for your license. As I recall, a few of my former instructors said that the FBI runs the check and makes out a report.

  That report had to be available to my state's board of nursing before it would issue me a license. I had a clean record, so I hardly needed to care about what the report might cover. You do need to learn if such a background check would include a past arrest, etc. Possibly a knowledeable lawyer can help you with that matter too.

  An acquaintance studying to become an X-ray tech had a problem resembling yours:  he had a DUI conviction. He wrote a state board asking if the conviction would bar him from becoming a tech. As I recall, the board responded that it would not automatically bar him from a license; it would consider his application.

  Consider one more matter. Suppose you manage to get a license as well as a job, and then a legal matter comes up, say a suit that involves you in some way with a patient. How will you fare then? I'm pretty cynical about suits and legal proceedings; I would bet the plaintiff and your own employer wouldn't hesitate to stick it to you if it served them do so.

Not all states require an FBI background check for employment and licensure.  Some only require a statewide check.  I've known people that had records and because they moved to another state those records were lost.  If you work in a state that requires a statewide background check, you have to have residence in that state for usually at least two years.  That information, even in a computer age is not shared state to state.  I would consult a lawyer to see what your best options are.