Search Api, fulltext and string for search bar and facets
I had an issue similar to this with 2 separate fields, Country and City.
Country was a taxonomy term while City was a textfield. I wanted to use both for fulltext search and as facets. In my Search API Index I added related fields for my Taxonomy (Country) and used the name (Country:name) as a fulltext field for my search bar. The thing is that the City when indexed as a fulltext didn't work in combination with the Country facet, and when I changed it to string it didn't work with the search bar. 

Solution: 


Leave your field (City) as a string field
Go to your Search API Index
Click on "Filters"
Check "Aggregated Fields"
Add new field and select your desired field, for example "City" and aggregation type "fulltext"
Go to your view, where you use the fulltext search, click on it and in the Searched fields you will see the aggregated field you had added before.


It should work fine now. If not, try re-indexing the items, clear your cache, double check that the field is set to be indexed, etc.