Sounds like you need to recaulk the thing, which is a lot of work (or money if you get someone else to do it). When doing so, do not sand twice! It causes a lot of extra work and you lose more of the teak than necessary that way. Remove the old caulking. Plenty instructions on the internet for it - spending on the special angled tool to scrape it out is worthwhile if you're doing the whole deck. Then recaulk, flattening the caulk with a putty knife (yes, smearing it all over the teak). After it's well and truly cured (a week at least, probably longer depending on temperature and humidity - see data sheets), you can then sand the deck until all the smeared caulking is gone, which means you have also smooth, nice teak left now. This way you only have to mask the edges of the teak, not every seam.
Use TDS-SIS 440 or any other MS polymer based deck caulking. Would advise against Sikaflex 290DC, as it requires primer and uses outdated PU chemistry, which costs more and gives worse results (poor UV resistance).