The caulking in my decks was also breaking down and tracking black around.
I didn't have a multi-tool when I tackled the cockpit seams so don't have a sense of what I was missing without it, but I did have good results with a bent screwdriver that came with the boat, and also the TDS reefing hook, which felt worth whatever I paid for it. (Can't remember what it was at the time, but it's currently way cheaper at defender than amazon.) I also cut the caulking away from the teak sides with a razor knife before reefing it out, which was key. The TDS seam sander took care of whatever was left over before cleaning and re-caulking with TDS caulk.While my teak was still thick enough to be worth re-caulking, it's not thick enough that I felt comfortable sanding it all down after ward, so I taped up all teak before re-caulking.