I would get this kitten in tomorrow if it were me, especially since this kitten already has signs of being sick; vomiting, diarrhea, drinking alot of water, and not wanting to eat.

It's not a good idea to allow a new cat to socalize with your current cat until the new one has been given a clean bill of health from the vet and is current on it's vaccinations.

I hope your new kitten gets better soon.