Does the bun with the gooey poop produce any round, normal poop at all?
If not, and certainly if the bunny is lethargic, it’s considered a veterinary emergency and he needs vet care. Baby buns get dehydrated quickly and can go downhill very fast. A vet can give sub q fluids and antibiotics. True diarrhea is caused by a parasite called coccidia, or overgrowth of bad bacteria in the gut.
If there’s normal poop as well as the gooey poop and the bun is acting perfectly normal, is lively and responsive and has a good appetite, this might be manageable at home. The butt should be cleaned with lukewarm water (not a full bath, just a butt bath) and dried carefully so it doesn’t run around with a wet butt and get cold. It’s likely the stress of being moved to a new home that brings this type of thing on. The bun should eat hay and the pellets he’s used to. No vegs yet.
Here’s an article from the HRS describing both true diarrhea and the much less worrisome condition called ISC, intermittent soft cecotropes:
https://rabbit.org/intermittent-soft-cecotropes-in-rabbits/