Answering "Yes, I do." to the question "Are you living in the county now?" is grammatically incorrect and, while still conveying the correct meaning, would sound awkward, even in an informal conversation.
A more grammatically appropriate answer would be "Yes, I am."
You can more easily see that this is correct by completing the thought that is being conveyed:
"Yes, I do living in the county now." (Incorrect)
"Yes, I am living in the county now." (Correct)
Another thing to note is that the contraction of "I'm" sounds awkward when used by itself.
"Yes, I'm." should be "Yes, I am," whereas if you explicitly state what/where you are, the contraction works: "Yes, I'm living in the county now."