Weird brown crusty stuff at the base of the tail.

Base, as in top of the tail? If so, either he's been biting at the skin there and it could be slightly infected, or he was recently neutered and it's something known as "stud tail". The latter is harmless and goes away a few weeks after a cat has been neutered.

However, you say he's not eating well and he's a new cat? Sounds like it's time for a vet visit.

He's likely fairly stressed right now and could benefit from you using feliway diffusers in the home.

I agree, a Vet visit if totally in order here. To check out this brown crusty stuff, and to discuss the poor appetite but drinking water.

Are you feeding the cat the same food he was eating where he previously resided?

Do you have other pets in your home that are stressing him?

Did you keep him isolated in a room by himself and slowly introduce him to your home/other pets/family members?

How old is he?

Does he have his own new litterbox?

Where did you adopt him from and do you know why he was up for adoption?

Well you should be contacting the Humane Society that you adopted him from and telling them about the brown crusty stuff, the poor appetite but drinking (drinking more than you might expect a cat to drink?), perhaps they will agree to have one of the Vets examine the kitty seeing how you're noticing issues not long after adopting. I wouldn't delay though, this kitty needs to be properly assessed by a Vet. Anytime I've adopted a cat from the Humane Society, despite their claims of 'good health', I still always take the cat to my own vet for a good check-up, that same day of adoption. My fear is that a Humane Society is so anxious to adopt out their animals due to so many cats and dogs coming in that they might not be as careful about screening for health issues.......and what is "no big deal" to them might actually be a deal. See a vet, and soon. A cat shouldn't be going for long without eating properly, it can impact their liver. If it's due to stress, that's one thing but if it's due to some underlying health issue (which is a common reason that cats end up at the Humane Society), that needs to be sorted out promptly or kitty could suffer or get worse.

Do you have other pets in your home?

Do you have any suspicions about the cat's history?

How did kitty seem at the H.S. at the time you adopted him; a lot more active and sociable but now very stressed out?

Is he pooping/peeing at all?

even though you're feeding him what he was fed there, who knows for sure if he was eating well THERE? In all honesty, they have so many animals there that I don't think they monitor these sorts of things that closely (which is no excuse but it's a reality). Try some smelly canned food..........fancy feast canned is one that even ill cats will eat, particularly the seafood varieties. Some cats hate canned food with chunks or gravy (some just like the gravy but won't eat the chunks). Try a can of the pate variety and a can of the chunks....try a few different kinds. Put some spoonfuls on a plate and see what kitty will eat.