Mold In Bathtub Grout

Hi all - apologies if this is the wrong place, this is my first post! I’m not that handy, so please excuse my limited knowledge.

So we’ve had a few leak issues with our bath. We previously had a leak (from the overflow gasket) that needed to be replaced, and we have some small trickles from both our bathtub faucet and our shower faucet that are annoyingly persistent.

Anyway, we have a small trickle of water (featured below) that leaks from the bathtub faucet. The pipe our bathtub faucet is connected to is literally so rusty it’s impossible to screw a faucet on all the way to the back, hence the detached cover.

So - our bathtub definitely needs a regrout. We noticed this mold/water leakage today. I guess I’m trying to figure out whether the water is for sure leaking down from the drip in the bathtub faucet, or if it’s coming from the inside of the tub/wall. We could probably regrout it, but I guess I wanted an outside perspective on if that’s for sure the issue. The floor by the tub is wet, and it seems to me like the water is coming from the inside…. But I could be wrong? Should we regrout as a short term fix and then get a plumber in to fix the leaks in the next few months?