EDIT: I guess I could import my 24 words + passphrase into Sparrow just long enough to sign a transaction back to an exchange right? I know entering your seed directly onto Sparrow probably isn't the safest thing to do, but if the plan is just to instantly send it off to an exchange to hold it until I get a new HW wallet then it's probably safe enough right?
Or, just use Sparrow on an offline machine to create a cold wallet and send your bitcoin there. Personally I would rather do that than trust a non-custodial exchange wallet, especially if we're talking about a significant amount. When you get your Cold Card back you can use that same seed to restore your Cold Card, and all your bitcoin will be accessible in your new hardware wallet. Since you created the new seed offline it'll be as safe and secure as if you had created a it with hardware wallet, and it'll save you a transaction.
Also, if you're sending the bitcoin from it's current location to a new seed, there's really no reason not to import the current seed into Sparrow. You would basically be turning your current seed into a hot wallet, but again, that will save you one transaction.
But how would I send from my current wallet to a new wallet if I don't have access to the cold card to sign the transaction? That's why I suggested importing the seed into sparrow to get access to it. If I just make a new wallet I can't send to it from my current cold card wallet to sign the transaction....