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 Google Domains subdomain to shorten links.
1. Go to Google Domains account.
2. Click Manage near the domain that you have already owned.
3. Choose DNS from the left.
4. Scroll down up to the "Custom resource records" block.
5. Enter the following values:
Name: your subdomain (in this example 'go'); Type: CNAME; TTL: 1H; Data: cname.short.io. Click Add.
6. Go to your Short.io account
7. Open the User menu and choose "Add a domain.
8. Choose the option - Add my own domain
9. Specify a domain and choose the second option. Then choose the subdomain from the list or choose 'Custom subdomain' and type in your value
10. Go to your domain settings to check if the domain is active. Refresh a domain status in 1-2 hours if the domain is still inactive.
Important: Wait a few hours and refresh a domain status in Short.io domain settings.
11. After the domain is active, feel free to shorten links and enjoy Short.io.