Good evening - I'm Dr. Sara. I'm a licensed veterinarian who works exclusively with dogs and cats. I'm sorry to hear that you're having a rough start with Rocky! I know it can be a bit scary when they're doing weird things like that.
It's concerning to me that you say he can't get up and walk on his own. There are such a tremendously wide variety of things that could be going on that I doubt I'll be able to narrow it down in this forum. He should be seen by a vet as soon as you're able - if not someone at the shelter where you adopted him, then your own family vet.
The timing is just about right for him to come down with something infectious that he may have been exposed to in the shelter. One big one that we get worried about would be parvovirus. Parvovirus is a virus that is very contagious and causes severe vomiting and diarrhea that can be life threatening. I have occasionally seen a parvo puppy who wasn't having the classic vomiting and diarrhea, but was just very lethargic and not eating and drinking, as you've described. They likely began his parvovirus vaccines, but their immunity isn't fully cemented until after they are 16 weeks old or so.
I'm reaching now, but I suppose he could be hypoglycemic. Puppies that have a low blood sugar can become very lethargic and "stumbly" or even have seizures. It wouldn't hurt to put some karo or corn syrup on his gums - it'll be absorbed this way and he will get at least a little boost of sugar.
He's just a little fella, and with what you're describing I am concerned about him. If it's possible to take him to an ER vet, that would definitely be the safest thing to do.
Please let me know what other questions I can answer for you :)