Hi, disney!
Well, I've definitely heard of allergic reactions to Novocain and the symptoms of it can be anything from hives and itching to symptoms of anaphylactic shock (which include the tight chest, difficulty breathing, etc). I have no doubt that the stomach ache
could also be a symptom, but it's not one I've read about (certainly doesn't mean it isn't also, though). I suppose you could also have a sensitivity to it (different from an allergy and usually categorized by nausea and stomach pains with medication sensitivities), or you might even have both.
Another possibility is that you might be nervous and stressed out when faced with dental work (lots of people are); the reactions you're describing sound a little like a mild panic attack or even the muscles relaxing AFTER your mind is sure that you're done and o.k. Obviously, if you don't feel like going to the dentist bothers you or you've had this kind of work, before, but never these issues, this might not have any basis at all, for you, but if you're one of many who reeeeally doesn't like it, this might explain it, as well.
I've also read a few things about allergies/sensitivities to the metal fillings in teeth, but don't really know enough to comment on the symptoms of that, so it might be something to look into.
Have you mentioned the feeling to your dentist or doctor? They might be able to help confirm an allergy or sensitivity for you and suggest an alternative or they might be able to help you figure out what's causing it.