I've seen some funky discrepancies with geo, but never in the 48% range.

It looks like you've already double checked to ensure your AdWords campaigns are only targeting people who are inside your targeted area and not people who are also interested in your targeted area.

If the traffic was all mobile, I'd say it could be an ISP issue. Have you segmented your data by Service Provider to see if the majority of the traffic coming from the out-of-state city is from a single ISP? Often mobile devices have IP addresses from large cities such as Chicago, Atlanta, etc. Maybe AdWords and Google Analytics don't always use the same data to determine location and AdWords could be using something more accurate such as GPS or search history while GA is using IP?

Your out-of-state traffic is still converting, which (hopefully) suggests it's not non-human traffic. Is your client getting phone calls from this other state as well? This would confirm humans are actually interacting with the ads and that the ads are serving out-of-state...