When Hester notices how feeble Dimmasdale has become, what does she intend to do?I thought she intended to tell Dimmesdale who Chillingworth really was, but I got it wrong.
You're correct that eventually she does decide to tell Dimmesdale the truth, but not yet. In Chapter 13 Hester is startled when she realizes how weak Dimmesdale has become. She realizes that keeping Chillingworth's idenity a secret is partly to blame for his condition, so she decides to go and speak directly with Chillingworth. In Chapter 14 Hester meets with Chillingworth. At first he tells her that people in the town have been discussing allowing her to take off the A, but Hester does not seem to care saying the letter will fall off when it's time.
Hester goes on to tell Chillingworth that he has punished Dimmesdale enough; however, Chillingworth doesn't see it this way. Chillingworh points out that he has not told his secret who had him killed (the true punishment of the time). In fact, in his effort to get his revenge, he as actually kept Dimmesdale alive. Knowing that Chilingworth is not going to leave Dimmesdale alone, Hester tells Chillingworth that the time has come; she is going to tell Dimmesdale the truth.