Q. Is this  a mistake on Wooster's part? 

A. No. It's your employer's mistake.  Your employer is required to withhold city tax for the work place location (Wooster in your case).  If they withhold the correct amount (the usual situation), you are not required to to file a Wooster return.  If  your employer does it wrong, you are required to file a Wooster return and pay the correct amount. Akron will give you a credit for what you paid Wooster, either by withhold or filing. 

"I was told I would get Wooster city tax credits". You probably misinterpreted that.  You have to pay Wooster tax.  Then Akron gives you a credit for what you paid Wooster.

You will need to pay Wooster, then file amended returns for Akron, to get refunds.

Since you essentially paid the correct TOTAL tax, just to the wrong cities, you might contact Wooster to see if there's any dispensation available. 

I found this with a Google search: https://about.usps.com/postal-bulletin/2017/pb22480/html/info_005.htm

Apparently USPS has not set up a withholding agreement with Wooster. I would think that means you're governed by the following: 

"In addition to the taxes listed in the table, there are other local taxes for which there are no agreements between the individual local taxing authorities and the U.S. Treasury Department. Employees must make their own payments to these taxing authorities. The Postal Service will not collect local taxes for a taxing entity that has not established an agreement with the U.S. Treasury Department."