What is the difference between a traditional chiro and a chiro trained in and practicing only the network spinal analysis method?

Very little. Network spinal analysis (nsa) is a marketing tool. Chiros function under an ancient, disproven notion of disease causation by spinal misalignment termed "subluxation" - something never demonstrated to exist. "treatment" is by forceful thrusts. Nsa postulates an improvement in function by gentle touch leading nerves to "retrain" for health. No evidence exists for this.
Stop seeing chiro. There are many things that could be causing your symptoms. If it is not a sore muscle then chiro (massage ) is unlikely to help. Heart pain, gallbladder pain, etc can give you similar symptoms and a chiro rubbing your back wont help u. See you md for a work up. If nothing serious is found you can go back to the chiro -- though if it hasnt hellped?