No, you do not have to amend federal or state if there is no change in tax as Tax Champ Mike9241 advises.
As you say, you should amend your local return. It’s odd your local wages would change but not your local tax withheld (Box 19). You may want to double-check with your employer about that.
If there is a change in local tax withheld, it may impact federal and state if you itemized deductions as Mike9241 says.
See How to Correct Federal Tax Returns
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