Ok, so it is not an ID, then you will still have to use the sets but can incorporate the single update as follows:

trigger AddCountyId on Contact (after insert, before update) {
	
//Create a unique list of Counties
    Set<String> CountyIds = new Set<String>();
    Set<String> AccountIds = new Set<String>();
    
    for(Contact c : Trigger.new) {       
       CountyIds.add(c.fdicCountyID__c);
       AccountIds.add(c.fdicAccountID__c); 
    }       
    
  	//Create a Map to match up Contact FDIC County Id and FDIC County Id
    Map<String, Id> countyMap = new Map<String,Id>();
    Map<String, Id> accountMap = new Map<String,Id>();
    Contact[] tbuCon = New Contact[]{};

if(Staticvariables.canIRun == true){

	//Add both sets of IDS to above Map
	for(County__c cty : [SELECT cty.Id, cty.fdicCountyID__c
		FROM County__c cty
		WHERE cty.fdicCountyID__c != null AND cty.fdicCountyID__c IN :CountyIds]) {
    		countyMap.put(cty.fdicCountyID__c, cty.Id);
	}
	
	for(Account acct : [SELECT acct.Id, acct.fdicAccountID__c
		FROM Account acct 
		WHERE acct.fdicAccountID__c != null AND acct.fdicAccountID__c IN :AccountIds]) {
			accountMap.put(acct.fdicAccountID__c, acct.Id);	
	}    
  
  	//Update Primary County on the Contact with County ID
        for(Contact updateC : Trigger.new) {
	    try {    
	          if(updateC.fdicCountyID__c != null || updateC.fdicAccountID__c != null) {
	  			  //Update Primary County with County Id from Map
	              updateC.Primary_County__c = countyMap.get(updateC.fdicCountyID__c);
	              updateC.AccountId = accountMap.get(updateC.fdicAccountID__c);
                      tbuCon.add(updateC);
	          }
	    }  
	    catch (Exception e) {
	   	}
	}

   if(!tbuCon.isEmpty()){
       Staticvariables.canIrun = false;
       update tbuCon;

   }

}
}

The static variable class would be something like:

public class StaticVariables{

Public Static Boolean canIRun = true;

}