Gum slicing pain from extraction a year ago. Recent Root canal, another extraction, causing more severe gum & tooth pain. 5 dentist & oral clueless.

Proper exam. you need to do your homework and find the right place to go to,an oral surgeon or a qualified dentist who can give you the proper physical and radiological examination to reach to a diagnosis and a treatment plan for your condition,weather it's a remaining root from previous extraction or remains infections I'm sure it can be taken care of.
Neurologist. Maybe a neurologist can shed some light on the cause of your continuing pain. Call for an appointment.
What type of tests? A lot could be going on, but what level of diagnostic testing has been done? Have you gone to specialists or to one of the oral medicine experts at the dental schools close to you or the hospital dental divisions where you live. You may have a multi- disciplinary issue. Start your search again with a fresh perspective! Good luck!
Possibly neurologic. May be neurologic rather than dental problem. Suggest that you be evaluated in oral pathology department at either Temple Dental or Penn Dental, both of which are relatively close to you.
Oral surgeon, then. See an oral surgeon. If problem still not solved, see orofacial-TMJ specialist dentist.