I am NJ resident but work in NY all year. I filed my 2021 taxes in April 22, according to which, I had refund from Fed and NJ and I owed tax to NY. Fed went through smoothly and NY gladly accepted the money I paid them.
Few months later NJ came back saying I had made an error and I owe them additional money instead of refund. I went back and and checked in TT and realized that NJ is correct (I had made some error in TT entries). I paid NJ the extra that I owed and now Fed and NJ are okay.
However, based on amended calculation its clear that NY owes me a refund plus whatever I originally paid.
Do I just have to send amended return to NY or do I have to send amended return to all 3 (Fed, NJ, NY)?
Is it required to explain why NY owes me a refund in a letter or just a simple amendment filing will resolve the issue?
If you discovered that your NY return was incorrect for any reason, you can file an amended NY return to pay additional tax or get an additional refund. Since you are a NJ resident, NY doesn't care about your NJ return, so only the NY return needs to be amended with NY. The amendment can follow the regular procedure for amending NY returns.
However, if your NY return changes, this may affect the credit you can claim on your NJ tax form for tax paid to NY. If it does, then you will need to file an amended NJ return to claim the correct credit.
You shouldn't need to amend your federal return. If you are deducting state taxes paid as an itemized deduction, you deduct those in the year actually paid, so the extra NJ tax bill will be a deduction on your 2023 federal tax if you just paid it (and you should report it, although you might not get any benefit because of the SALT limitation).