Hmmm.... What I would do and have done in this case is first, get a sharp tool like a wood chisle and a box knife. Dig out all the old stuff. This is the hardest part. I try to get it all out for a nice clean new bead of caulking.
Once the stuff is out to your satisfaction, I would spray the area with bleach or vinegar and let it sit for a day or so to kill the mold and algea.
Once dry, then I would do my caulking job.
Caulking is easy but you need some Caulking 101.
Do not cut the nozzle where the hole is too big, this is the biggest problem most have. Cut the nozzle with a sharp box knife and make it an an angle from 45 to 30.
I caulk in the direction that allows me to see the caulking bead to decide if I need more caulk or if too much is coming out. You want to first press the angle nozzle hard to the corner to get the caulk into the crack. then the slow steady movement will create the nice clean angle that will allow water to roll over the bead and into the tub.
When all caulked, then dip your finger in some clean water and move it in one direction over the bead just to smooth it out. In your case I would use 100% silicon caulk in pure while.
Now, the drying time, I would let the caulk dry for one 24 hour span and during that span not allow any liquids, water or chemicals to come in contact with it.