Moist dermatitis is a bacterial infection. It's a very specific one that I posted a link to above, that causes the fur to turn green. Your rabbits paws are getting dirty too because she is trying to wipe and clean her face, so the drool and dirtiness is getting transferred to her paws.
You saw the drool coming out of the mouth. The vet really needs to do a thorough dental exam, where they don't just check the front teeth, but use a scope and light to check the back molars and inside the back of the mouth. It could be the molars have overgrown and now have sharp points cutting into the inside of the cheek and tongue, or there may just be a piece of food stuck back there in the teeth irritating the mouth. Or you may need to have xrays done of the teeth, as this could be happening due to an infection. But it's important for the vet to investigate further, because this isn't just going to stop and go away on it's own.
Since your vet isn't an experienced rabbit vet, you will need to be insistent with the vet on investigating the cause of the drooling. Trimming the fur and wiping the area, may help a little in keeping it cleaner, but it's not going to solve the underlying problem of why the fur is getting wet and why your rabbit is drooling. This will keep happening and will get worse, even to the point your rabbit may stop eating. So tell your vet you saw your rabbit drooling, ask them to look inside the mouth at the back teeth, ask for xrays if you think that needs to be done. I would also request that the vet does the fur trimming while you are there. They have electric clippers and it should be much easier for them to trim the fur than you.
Because your vet isn't rabbit experienced, if they want to administer or prescribe any medications, please double check that the medications are considered safe for rabbits. Some medications, antibiotics in particular, can be deadly to rabbits if given to them. Also, because your vet isn't rabbit experienced, you will want to ensure they handle your rabbit in a safe manner, so no accidental injuries occur, as rabbits are very fragile creatures and can sustain injuries easily if not handled properly.
Medirabbit (rabbit safe medications)
Medirabbit (harmful medications for rabbits)