here is an example of a CX bike with some very aggressive tires. You don't need this extreme of a tire for what you described.
Something like this is what you're looking for in my opinion. As previously stated, it's your old 10speed bike with wider, more grippy tires. Tires provide a tremendous amount of comfort or discomfort. These larger CX tires allow you to run a much lower PSI vs road bikes. This bike you can scoot around on the road quickly, will have sufficient grip in a dirt or gravel road and your chipped roads will be perfect for this type of bike.
