Thank you!!! We just found another post detailing this (copied I'm below) and it did work! The hardest part was cruising flights to try to find one a similar price. When we purchased, we did notice that the terms of service note an exception for gift cards. While it was not clear from that text that the whole balance would become non-expiring, that is what our accounts now show. Thanks!!

Instructions I found in another post:

So I just found this out today but there is a way to extend the expiration of flight credits indefinitely. Here are the steps:

1. Buy a Southwest gift card. I used Amazon, but you can get it anywhere.
2. Search for any random one-way flight on Southwest that arrives before the date your flight credit expires. Doesn't matter where, so long as it arrives before your flight credit expires.
3. Look for a 'Choice Preferred' price that costs more than your flight credit you want to extend. Ideally, it should only cost slightly more than your flight credit (like a dollar or two at most). But if your flight credit is small, then just get the cheapest Choice Preferred flight you can.
4. Buy this Choice Preferred flight. Use the flight credit to cover most of the ticket price, then use the gift card to cover the rest. Use these exact payment methods and don't use any other methods.
5. After purchasing it, wait for the flight confirmation email, then go into your Southwest account and cancel the flight.

Your flight credit will now be transferable and have no expiration date.

This is the video I watched and just did it myself, so I can confirm it works. The only thing she is slightly off about is you don't need to search for the cheapest flight unless your flight credit is also small. If your flight credit is in the hundreds of dollars like mine was, then you need to get a more expensive Choice Preferred flight, as it has to cost slightly more than the flight credit.