If the question is "Would you care for a glass of wine, the answer would be, "Yes, I would."
If the question is, "Would you like to have a glass of wine?" the short answer is, "Yes, I would," or "Yes, I'd like TO."
If the question is, "Would you care to have a glass of wine?" the short answer is, "Yes, I would." We just don't usually say, "I'd care to."
I'm still having trouble understanding what you say. This is the original unit:
A: Wouldn't you care for just a glass of wine?
B: __________, but I'm the designated driver tonight.
According to your explanation above, the right response is "Yes, I would" not "Yes, I'd like to". But you originally said it must be "I'd like to". Could you tell me what I misunderstand?