Hi there - I'm Dr. Sara. Congratulations on your new adoption - although I'm sorry to hear that he isn't feeling well.
The stress of sheltering or rescue and a move to a new home can definitely trigger some GI upset, including diarrhea and possibly vomiting. Depending on the type of situation he came from, he could also have picked something up with his shelter or rescue. Any time you get a bunch of animals together, they all pass around their germs - just like kids at the start of school. It's not unusual for a dog to get diarrhea or have a bit of a dip in appetite when they come to a new home.
In general, as long as they are acting normally (I know that you don't know him that well so this can be tough to define) and their gums are nice and pink and moist, I don't get too concerned. If they are acting very lethargic, though, and unable to keep anything down for more than half a day or so, then I get worried that they may need veterinary care.
If you have any other dogs, I would suggest keeping them separated just in case your new boy has something contagious.
With diarrhea it's reasonable to offer what we call a bland diet. This is a diet that consists of 3 parts boiled white rice to one part boiled chicken or turkey breast or ground beef with the grease rinsed out. Since he just vomited, I would wait a few hours before you offer anything else, and start with a little bit of water or ice cubes.
If he is very lethargic, his gums are pale or very dry or he is continuing to vomit, I would seek veterinary care. Sometimes you can also call the organization where you adopted him for more advice as well.
Please let me know if I can answer any other specific questions for you.
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