Employer mistakenly withheld NY taxes instead of NJ

I am a NJ resident.

My company is based in NJ , some client offices are in NYC.

This March we were supposed to move to NYC client offices but due to the ongoing pandemic it never happened.

I am working from home since mid-March 2020.

In November I've noticed that my employer withholds NY taxes from my paycheck since March.

Apparently , they received an update about our move to NYC and updated payroll accordingly.

No one contacted me to verify the move.

When I raised the issue with my company, they said that payroll stubs cannot be corrected retroactively.

They then said that I will get my money back when I file taxes.

They refused to provide an official letter explaining the situation.

Several questions:

 1.  Should I insist on an official letter for IRS? I never lived or worked in NY , yet taxes were paid to NY. 

 2.  Will I be able to file NY non resident tax return while claiming $0 income in NY state ?

 3. I read about similar cases and general advice is to file NY non-resident first, then NJ resident. Is this correct in my case?

Any other things I need to do/be aware of  to get my money back ( NY taxes are higher than NJ)

Thanks in advance