The Addressfield module stores an abbrivated name for countries and state such as US for United States and FL for Florida. The Search API indexes the these words which are not very useful when doing searches for places like New York, Boston, France, ect.. Should I just discard the Addressfield module and make text fields or is there another way to go about doing this?
you might want to use SOLR search and write a synonym.txt file to translate the state and country codes into names
Unless somebody is going to write a rather sophisticated plugin or module to do this as contrib to either the Addressfield module or Search API I doubt this will ever work. The Addressfield module is extremely complex not opening itself to easy integration. Sadly, the best solution is the simplest which is to create text fields. The Geocode module still geocodes the addresses which can be used by the Apache Solr Seach API which is nice.