There is a form that can be used to change an address with the IRS. It is Form 8822. Here is a link for further information:
Topic No. 157 Change Your Address -- How to Notify the IRS
There may be a similar form available for your state, or you could contact the state department of revenue for advice about the situation. The following TurboTax article has a link to each state's department of revenue:
How do I contact my state Department of Revenue?
You do not need to file an amended return for Federal or state to correct the zip code.
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