Drooling is absolutely a possible side effect of amoxicillin. Also look for lack of appetite, diarrhea and vomiting. Signs of liver disfunction, like jaundice of the eyes, skin etc. are less likely but also possible. There are also more severe symptoms of anaphylactic shock (allergic reaction) that could occur if you continue giving the medication.
The best course of action is to discontinue the amoxicillin and take your kitty in to the vet as soon as possible.
Welcome to the forum. Let us know what happens. Good luck.
Was it in liquid form? My cats do this when I have to squirt it in their mouths. I think it's just the saliva. They usually don't like the liquid medication. I always ask for pill form. If you see that your cat is having trouble breathing or looks like it's face is swollen then you need to get it in right away to an ER vet so that they can give it a benadryl shot and monitor your cat. Don't wait if you truly think it's an allergic reaction. Watch and see how he/she is acting. Let us know what happens.
Best wishes.