Unfortunately in a senior cat there are several possible causes for what you are seeing. Problem with the kidneys, liver or blood glucose levels(diabetes) can all cause the symptoms you are seeing. Not doing blood work is unacceptable in a senior cat and I would advise seeing another vet right away as all of the things listed above can be managed with medication. Diabetes and kidney disease is very common in senior cats so it has to be considered.
Vestibular disease is also possible. This is often caused by a problem in the inner ear, however we normally see things like rapid eye movement with this as well.
Here is additional information on this
In a senior cat a brain tumor though rare in cats also has to be considered.
Here is additional information on brain tumors
Because this may be related to the kidneys or diabetes I would have him seen by another vet as soon as possible. It would not be a bad idea to take him to your local Veterinary emergency if you are unable to find another regular vet open today.
Until blood work is done there is no way to know what is going on and what treatment is necessary.
Until you can get him in Pepcid AC may help stop the vomiting. !/4 of a 10mg tablet every 12 hours. Wait 45 minutes after the Pepcid and offer a small amount of chicken baby food( be sure it has no onion or garlic) and see if he is able to hold this down.
Please let me know how things are going.If you need help locating a ER in your area please provide your zip code. I hope another vet will be more helpful in finding out what is going on with your boy.
Please reply back if you have additional questions