My gums hurt. Does this sound like herpangina? There aren't actual visible sores.

No. Herpangina is a very specific clinical syndrome with coxackievirus infection producing tiny blister-like lesions on the back of the throat and on the fauces (the tissue ridges that meet in the center and house the uvula).
No. No... If you are an adult, it sounds like a gum issue. Please see a dentist for a comprehensive exam, including gum health and gum pocket measurements.