You can choose "Single Address" or "Multiple Addresses":

How do Bitcoin autowithdrawals work? https://help.swanbitcoin.com/hc/en-us/articles/1500010403461

Multiple Addresses require that you share your cold storage's xPub with Swan:

How do I connect my wallet's extended public key (xpub/ypub/zpub) to my Swan account? https://help.swanbitcoin.com/hc/en-us/articles/360060455013-How-do-I-connect-my-wallet-s-extended-public-key-xpub-ypub-zpub-to-my-Swan-account-

Of course, that means Swan can see your balance and know of other transactions to any addresses using that xPub.

To prevent this, with ColdCard, you can set an "account number" (a custom derivation path). You need to keep a record of this "account number" / custom derivation path (e.g., keep it along with your seed backup), otherwise I don't think you'll have success trying to recover without it).

Derivation Paths and Accounts https://coldcard.com/docs/paths

Also, remember that every "automatic sweep" creates a UTXO on your cold storage. Sure, you can spend those cheaply today. But who knows N years in the future what the fee per UTXO will run.