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 Namecheap subdomain to shorten links.
1. Go to the Namecheap account to the list of domains.
2. Select "Manage" next to the necessary domain.
3. Click on the "Advanced DNS" tab.
4. Add "A-record." Input your subdomain to the HOST field. Add 126.96.36.199 VALUE (an IP address, which connects Short.cm and Namecheap).
5. Go to the Short.cm account. Click the "+" icon.
6. Choose the third option — Easy use. Click "Use existing."
7. Enter a domain you already use.
8. Input a subdomain to the "Other" field.
9. Click "Complete."
Important: Wait a few hours and then refresh the domain status.
The video instruction helps to configure a Namecheap subdomain.