It's not just the fare class but the fare basis - more characters - that determines what you can do. Some nonrefundables allow changes to the dates of travel but not to the cities. As noted, you need to call CO to see what you can do. Call and say you want to make a change and have ready the flights you want for the new ticket. Do not say you want a refund, just a change. With any luck you will get what was described above - your new ticket and a credit for the balance. BTW you can expect to have to pay the change fee with "new money" and not deducted from the already paid fare.
As always, if you get an agent that says you cannot do this say thanks, hang up and call back to get another agent. Repeat this as needed.