They can't hold an open balance over your head. You may cancel the card and so long as you continue making monthly payments at or above the minimum due and pay the balance before the offer expires, you will owe no interest.
As you know, using your card for any other purchase requires you to pay off in full or be hit with an interest charge. You were right not to use that card. Despite that care, the annual fee got you. Call Chase and protest the "purchase interest fee," not because 33¢ is a lot of money but to ensure you're not going to be hit with a substantial amount next month!
By the way, this is not a violation of the pay-the-highest-rate-balance first rule, because the annual fee charge triggered the low-rate balance to flow back into the high-rate balance. Thus, there is just one balance being paid off.
I called and protested, and the CSR was understanding and apologetic that the previous rep I spoke with seemed to "miscommunicate" with me. He assured me there would be no further interest charges, and credited my account the $0.33. lol, gee thanks.
So, at this point, SM retention/loyalty or call? I just paid Chase the AF on 3 cards...would love to get something back. I'm new to Chase retention negotiation, though...but I assume it's close to Citi.