Yes
It will ding your credit score, even though you don't want additional credit. I know, it's ridiculous, but that's the way it is. The banks don't seem to have a mechanism to apply for a different card without applying for additional credit. And once the application is sent, there's a Hard Pull on your credit.
You could always ask the bank to 'convert' your card to a different account, which won't harm your credit at all. But then there are no signup bonuses either.