I'm sorry for your misery, as I've had a similar problem with one right upper molar which many dentists found was a perfectly healthy tooth. The pain would shoot up to my eye and brain and felt precisely as you described,like someone pulling my nerves out. The pain also fluttered, like changing, I imagine, branches, like a tree. I imagine the pain travels through the branching networks of nerves, so that's what may be going on with you - the pain traveling through to other areas. Like lightning in the sky.
Since I had no infection, a dentist finally decided to just drill a hole in the tooth to see if maybe pressure had something to do with it. Well, the pain immediately subsided and did not come back, still trying to figure that out, until I had to remove the tooth due to an accident decades later.
Now, if yours is indeed an infection problem, the following might be able to help you with the pain and swelling until you can resolve the source problem.
My sister had an abcessed molar that swelled her gums, lymph nodes and the whole side of her cheek down to the neck, but was unemployed and could not afford a dentist.
I suggested, based on my many experiences with infections, to rub a good amount of CASTOR OIL on cheek and neck, and place a hot water bottle for heat, as heat makes the oil penetrate better. At night she would just apply the oil before bedtime. She did this for three days, the first day alone diminishing her pain and swelling substantially, and by the 3rd day you could not tell any swelling and she felt almost normal. It stayed this way until she was able to go to the dentist. I felt she should have continued to get even better results until then, but she was "satisfied" with just that much.
Scientifically, Castor Oil has been found to contain an element that causes the creation of immune cells that fight infection (I tried to find the article for you but its taking me more time).
I have used Castor Oil for all infections and have had excellent results, the greatest being completely dissolving a cyst the size of a golf ball that grew behind my navel, from an operation via my navel to tie my fallopian tubes; and earlier an uterus infection that would not take any Antibiotics for its cure for a decade. Its good for: any infection, bug bites, spider bites, ear infections, conjunctivitis, cuts, surgery surture sites, bruising, Arthritis swelling, knee pain/swelling.
It is also used to gently detoxify your liver, pancreas, gull bladder, appendix, spleen, intestines, colon...by applying a Castor Oil Pack (oil drenched 2 fold sheet of wool flannel or cotton spread over the areas) over the areas and abdomen, then a plastic sheet over that and then a heating pad over that for 1 hour and a half, for 3 consecutive days - then take a tablespoon of Olive Oil. This is what I did for the uterus infection. Was cured in 3 days - never more an infection.
Something else you can add to that.... look up Jon Barron and his baselinenutritionals. He has a phi zymes, look that up,supplement that cleans your blood and many testimonials are coming in that people's dentists are reporting to them that they have NO plaque on their teeth - which was an unexpected result. There's much info on his site and it is searchable. That could possibly reach the underlying problem in your teeth.
Also look up biological dentists - they use supplements and homeopathy in their practice. Their approach to dentistry I would say is more leading-edge as wel as holistic. I know an excellent one in upper NY and several in NJ. Many use energy medicine.
I sure hope any of these help! It was sharing from others that lead me to help my self when medicine gave me up for hypochondria. Be well!...