I would suggest removing tho old caulking completely as it sounds like its beginning to reach the end of its life. then scrape the old finish off the teak, amd re caulk. You can make a neat job of it by using masking tape on the sides of each seam. Its work,but really not that bad. Because its a teak veneer, i would be a little nervous about sanding. I would then recommend no finish, and definitely no oil because all it does is attract dirt, mold, and algae. To clean the bare teak, scrub with salt water and a medium stiff brush. If you are in fresh water, buy a box of pickling salt for about a buck, and make your own.