To answer your question first, yes, it is possible to have damage to the nerves and blood vessels of the teeth that isn’t because of tooth decay.
Possible causes are many, ranging from cracked teeth, infection from surrounding structures, congenital conditions such as dens invaginatus.
To help figure out which is the problematic tooth, there are various tests that can be done, such as electric pulp tests, cold tests, percussion and palladium tests, x-rays, cone beam 3d tomography etc.
Sometimes, however, diagnosis can be challenging and further investigations and time is needed. It’s also possible that other issues, such as a sinus infection for example, may be causing referred pain to the tooth.
Thus, it’s best to see a dentist you trust to help you to figure out what’s going on so that appropriate treatment can be prescribed. All the best!