Lets be clear on few things,
  
  You cannot get callout response in trigger in realtime, since it is async response will be asycnhronous.That means in the future method  you have to write a logic to read the response and based on that you can update your account email address field. If it errors out you can send an email or create a new field on account to store the response if you want. 
  
  Here is the sample.
  
trigger EmailChange on Account (before update) {
    for(Account a : trigger.new){
      if( Trigger.oldMap.get( a.Id ).Email__c != Trigger.newMap.get( a.Id ).Email__c ){
         //You cannot get response here, it is async so you cannot read response 
         string APIResponse = APIConnect.makePostCallout('email',Trigger.newMap.get( a.Id ).Email__c,a.Id);
         //a.addError(APIResponse);
         // add more functionality here
         System.debug(APIResponse);
      }
    }
}

public class APIConnect {
    @future(callout=true)
    public static string makePostCallout(string mode, string var,String accountID) {
        Http http = new Http();
        HttpRequest request = new HttpRequest();
        request.setEndpoint('https://myserver.com/sftest/');
        request.setMethod('POST');
        request.setHeader('Content-Type', 'application/json;charset=UTF-8');
        request.setBody('{"key":"3434545","mode":"'+mode+'","var":"'+var+'"}');
        HttpResponse response = http.send(request);
        //Read response.
        if(sucesfulresponse)
         {
           //Update email address on account 
           Account accRecord  = new Account(id= accountid, Email__c = var);
           update accRecord;
         }
        else
        {
          //Error
          
          //you can send an email
          
          //Or ceatea new field to store the response.
           Account accRecord  = new Account(id= accountid, newfield = response.getBody());
           update accRecord;
          
        }
        
    }        
}