Welcome! My name isXXXXX am a 2003 graduate from UC Davis, and currently a Medical Director of a veterinary hospital.
I am sorry to hear about this concern.
Well, it certainly sounds a lot like a bad vaccine reaction
Yes, you can give benadryl.
Benadryl can be given at a dose of 1mg per pound of body weight. Keep in mind, Benadryl tablets and liquids come in different sizes. So, an approximately 25 pound dog can get a full 25mg tablet or a half of a 50mg tablet. I usually avoid the liquid Benadryl in larger dogs (you would have to give too much of it). You can give Benadryl every 8-12 hours.
Is it enough?
Generally, with prominent swelling, I do strongly recommend an emergency vet visit. They can give a steroid shot that can quickly help to reduce reaction and swelling. That would be the best course of action in addition to also giving benadryl ASAP.
I hope that information has been helpful.
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