Yes, in this example both are correct. Use "I had to" if the MAIN purpose of the sentence is to describe your conversation with the doctor. "How was your doctor's visit?" "OK, but he only cares about my physical health, not the stuff that really matters. He said I *had* to get rid of my cat, but there's no way I'm going to do it. I love that animal, and I will never get rid of it, and the doctor is a jerk if he doesn't realize that." Use "I have to" if the MAIN purpose of the sentence is to discuss what you will do with your cat. "My doctor said I *have* to get rid of my cat. Do you know anyone who would be willing to adopt it?"

When you say "I have to do something about my cat " I interpret that as you need to do something about your cat in the near future. That action hasn't been done yet. When you say "I had to do something about my cat" I interpret that as you had already done or refused to have done something about your cat, which prompts the question "What did you do to it?"