Trying to map postal codes to their respective cities, but running into some issues

The Problem

I am creating a database of Countries, Regions, Cities and Postal Codes. Here's an example to give an idea of the database structure as it is now:

enter image description here

Now, let's take Tuskegee University as an example to illustrate my problem. Here is Tuskegee University's address:

1200 W Montgomery Rd
Tuskegee, AL 36088

For the city database, I grabbed postal code data from Geonames, and I cross referenced this data with USPS. In both dumps, for '36088' I get Tuskegee Institute. So, when I do the following:

SELECT a1.name,a1.address,b1.name,b1.postal_code    FROM university a1    JOIN city b1 ON a1.city_id = b1.city_id   WHERE a1.name = 'Tuskegee University';

I get back:

"Tuskegee University";"Kresge Center, 3rd Floor";"Tuskegee Institute";"36088"

Thus, making my address:

Kresge Center, 3rd Floor
Tuskegee Institute, AL 36088

The Question

There are many problems here (e.g., the address: I downloaded university data from IPEDS and obviously their address does not match Google Places). However, what I want to focus on is the issue with my city table. I feel like the schema here may not be the best approach to the problem, but I'm not really sure what the best approach might be. Do I create separate city and postal code tables? But, in the end, how do I associate 36088 with Tuskegee, AL? Is there a better source here that I have just not found?

A Possible Solution (with problems)

Say I switched up the DB schema, such that it looked like this:

enter image description here

Then, I could have the postal code and city be independent of each other. Thus, the university table points to Tuskegee for city_id, and the 36088 for postal code. However, I now have an issue. If someone types in a city, I cannot tell them cities within a certain radius. I could get the lat/long for cities and create a geog column in the city table as well, but that seems a bad solution. I could associate cities with postal codes (i.e., city_id column in postal_codes table), but then I am faced with the same problem of mapping 36088 to Tuskegee, or having 'Tuskegee Institute' in my city table (which I obviously don't want).