Hi there @Ksohrab!
Thanks for outlining your query so clearly for us here in the Shopify Community.
Would you have previously used this domain in conjunction with an earlier Shopify store you had opened?
If so, and if you have proof of ownership of this domain you can reach out for account support via our Help Centre here where we'll be able to verify your identity and domain ownership.
There we will be able to check this out with you securely and free up that domain for you so you can add it to your new store.
Can you tell me a bit more about your business and what you're selling?
What else are you working on for your store at the moment, are you building up towards launch still?
If a site is available with the same name then you can't register it. It better to add one alphabetic in that domain then its become fine for you.