Depends on how original posts were installed. If they do not have concrete inside the bottoms an inner sleeve works.Good chance the posts are empty below grade if concrete was placed after setting posts in place. Not my first rodeo.The weak point (if any) is right where the existing post meets the concrete curb (mow strip).
Therefore, anything added to the inside of the existing posts will not help strengthen the posts.
Cap the new posts to keep water out.
Bill has a good plan.