I apologize, I answered your second question and had to go back and read the first question since they are two different queries and concerns of yours. I do not see the issue as far as how SLP is working, You introduced a new issue when you referred to typing in the name differently , as opposed to not seeing results when a correct city was typed in with no zip code
When I typed in Walker, Minnesota I saw two results, it took a few seconds to load though, you must have a lot of locations , see screenshot:.
How Google recognizes whether it is a city name or a state abbreviation shouldn’t matter but sometimes they want more info and unless you have turned on show address guess or some other SLP Premier search enhancements we have no control over what is returned by Google under WPSLP and I do not see any issues with how SLP is working on your site.
When I typed in just the state Minnesota into search box it returned two locations.:
Mills Detail & Trailer Sales
1916 Albert Avenue South
Park Rapids, MN 56470
United States
Directions
L & M Fleet Supply
1307 1st Street East
Park Rapids, MN 56470
United States
Directions Website