Nominally, the customer table should only contain the street address and zip code ID (not the city or state). Note that data entry for such a normalized scheme is not necessarily straight-forward; people will expect to enter city, state, zip code (or maybe just zip code) and the onus will be on the application to map that correctly and disambiguate when necessary.
I too live in a zip code used by two cities; they even happen to be in different counties, which leads to questions about which county I live in on occasion. One of the cities has multiple other zip codes; the other only has (part of) the one zip code. There's no problem: you would have two separate ZipCodeID entries for the same ZipCode, one for each city. Note that this means that there would not be a unique index on the ZipCode column in the ZipCode table.
Where would you store the +4 of the Zip+4 scheme? Good question! That belongs with the street address.