Recaulking the shower and found some gaps. Help?

Started recaulking the shower because of mold/ ceiling was soft underneath it.

Im having troubles cleaning it and figuring out a plan for the gaps.

I don’t want to landlord special it, so here was my plan if anyone wouldn’t mind correcting me!

- find a way to get rid of the gray stuff that doesn’t seem to be caulk (I’ve tried a steel wool brush, kitchen brush, scouring pads, with vinegar, hydrogen peroxide, and bleach on separate occasions.)

- install foam backer rods into gaps

- grout & seal over gaps (that aren’t in corners/changing materials) and caulk

Thanks everyone!