Hi. The proper caulk to use on the roof is "self leveling caulk". Lap sealant is used for vertical surfaces, ie around Windows, along exterior wall trim, etc.
On my roof, I installed Eternabond tape along the front and rear seams only. The rest is self leveling caulk.
I don't use Eternabond tape around roof fittings such as vents because in the event I ever need to remove or replace the vent fixtures, removal of the tape (once applied) is very difficult.
The best thing to do is clean all existing caulk areas on your roof with a hand scrub brush to remove all blackened areas. You can then better assess what caulk needs to be redone. Once the existing caulk is good and clean/dry, you can apply more self leveling caulk right on top of it.