You can use a "plus alias" of the same email so it frees up the original address without needing a second inbox.
Sign in to the old account, click your name or profile icon in the top left, and choose Profile.
Click the email field and enter a plus-alias of your current address — in Gmail, add a + and any text after your username, e.g. example+oldaccount@gmail.com. Mail still lands in your normal inbox.
Click Update.
If the old account has a password, you'll be prompted to enter it to confirm the change. Accounts using magic-link (passwordless) sign-in won't be asked for one.
A confirmation link is sent to the alias address (which, since it's an alias, still lands in your normal inbox) — click it to complete the change. The old email stays active as the login until you confirm, and the old address also gets a notice that the change is happening.
Once confirmed, your real email address is free to use for signing up a new Thatch account.
If you're still stuck (can't access either inbox, or the plus-alias trick doesn't apply to your case): contact Thatch support — they can change the email directly from the admin side, though that skips the confirmation step and you won't get a notice, so it's reserved for exceptional cases.