I'm not sure how the last comment was a solution. It's missing key information. When I performed the following steps, I was able to start shortening URLs with my GoDaddy domain. It shows up on the shortener page and properly redirects to any URL I want, and adds an entry to my list of shortened URLs, just like x.co and go2l.ink used to do.
First, you need to edit your "A" record. This can be found by going to dcc.godaddy.com and clicking the three dots to the right of the domain name that you want to set up for URL shortening. Select "Manage DNS" on the dropdown.
You should now see the records list, with the entry for the "A" record on the first row. Edit that by clicking the pencil at the right side. Enter 126.96.36.199 into the box labeled "Points to" and ensure the "Host" box has the @ symbol in it. Click the save button.
Next, you need to edit the TXT record. I did not have one, so I had to click the ADD button that's at the bottom of the list of records. For the Host, enter the @ symbol. For the TXT value, enter your 8-digit ShopperID number. Click Save.
After that, you need to scroll down right under the records section, to the forwarding section. Click on ADD to the right of the word "DOMAIN" and you'll see a URL box entitled "FORWARD TO" -- enter 188.8.131.52 into this box and click the save button.
That's it! It started working immediately for me.