When I first got Duke I had similar issues. He was a little underweight and was not interested in his food at all. He would eat it out of my hand but otherwise he had no interest in the bowl and even if I could coax him to start eating out of the bowl, I would have to stay in the room or else he would follow me. I tried wet food with similar results, runny stool and a couple of accidents in the house (he was potty trained, you could tell it just upset his stomach and ooops out it came; is yours potty trained for sure or is he still confused about where is appropriate to go?). Duke also had a few "vomitting" occassions where he would eat and a couple of hours later he would throw it back up pretty much solid. That made me worry about a blockage, so I of course called my vet (which I suggest you do). It never required a check up because it wasn't a blockage, it was probably just a mixture of stress/anxiety, but it was still great to have the vet aware of the situation so she could tell me when it became a very big issue (blockage is a huge issue).
Duke never lost any of his energy throughout this and drank water with no problems, which was a good sign. To help him get his appetite, instead of using wet food which had negative poop results, I would sprinkle some cheese (shredded or parmesan) to make it more appetizing. I would also sometimes mix peanut butter with water and microwave it and pour that warm mixture over his dry food. The warmth and wetness make it more appealing/easier to digest. My vet also recommended adding canned pumpkin to his food because that hardens a dog's stool. I would definitely have a conversation with your vet about that before using it as your solution, though, just in case there are other issues at hand (worms, blockage, etc).
Introducing a dog to a new environment can be extremely stressful, and that sounds like what may be causing some of your dogs issues. I would definitely call your vet (its free to call!) and get some advice to follow and if he doesn't get better or she suggests a visit, go seek professional help before things get worse. Otherwise, as suggested definitely follow the two week shut down. I didn't do this right away with Duke and it would have helped speed up the "return to normal" process for him had I done it sooner! Think about the world from his perspective, going from one home to a kennel to another home is extremely stressful especially since he is young enough all that happened during his socialization/learning stages (same as Duke, he started life neglected/abused and ended up in a kennel for his toddler to teen stages... not a good intro to how to act as a dog!).
Thank you so much for saving your precious new family member. I hope he feels better soon!