Turbo Tax keeps going back to my PA return, which is complete. How do I proceed?

Are you receiving a specific error message? Please advise so we can better assist. 

OIssue seems to be resolved.  Question now is Why am I being asked to file a NYS return when I don't owe the state anything?

The question here is whether you have any New York source income or if you are a New York resident. 

I am going to assume you are not a NY resident. If you were, you would likely have to file a NY return. 

If you do not have any NY source income, I would check the entries on your W2s to make sure that NY was not added to one of them, both in the program and out. If you do have NY source income, you would likely have to file a return for NY even if you have no balance due.