If you go to: https://snapcraft.io/YOUR-SNAP/listing and you scroll down to Contact information, you can verify domains such as your primary domain by setting a TXT record in the domain names DNS records and waiting up to 24 hours.
The text record needs to contain something like: SNAPCRAFT_IO_VERIFICATION=123abc123
Where I went wrong, and the tip I would give you, is that the TXT record has to be set at the root of the domain (@) and contain the exact format above (key=value). I had incorrectly set it at snapcraft_io_verification and then set the value to be 123abc123.
Hope this helps someone!