It is not necessary to register a new domain name for shortening links if you own one that works properly. Create a subdomain using a domain you already have.
Learn the instruction on how to add and configure a GoDaddy subdomain to shorten links.
1. Go to the GoDaddy account to the "My products" page.
2. Select "DNS" next to the needed domain.
3. Add the "A" record. Input your subdomain to the HOST field. Add 188.8.131.52 VALUE (an IP address, which connects Short.cm and GoDaddy).
4. Go to the Short.cm account.Click the "+" icon.
5. Choose the third option — Easy use. Click "Use existing."
6. Enter a domain you already use.
7. Input a subdomain to the "Other" field.
8. Click "Complete."
Important: Wait a few hours and then refresh the domain status.
The video guide helps to configure a GoDaddy subdomain.