Yes it's normal to get zingers in the teeth as the nerves run into each tooth unfortunately.
That is why some people confuse TN with dental issues, going so far as to even pull teeth out unneccesarily. I've actually checked to see if a tooth is loose, even knowing it's not, it just felt that way. It can move from tooth to tooth, zing, ache, burn, all differnt sensations, including gums, lips, the works. Jazzer1, sorry that this terrible ailment would happen to a musician like yourself.
I am the sax and woodwind player. Glad to have heard from another woodwind player. I also get pain in my teeth as you describe, also jaw, toungue, lip, face. I am going to try getting as close to eliminating caffeine, chocolate, nuts, and saturated fats as I can to see if diet helps, as I have heard that it does. My job in the West Point Band's Jazz Knights for 14 of my 23 years in the Army was performing baritone sax. I was in a great deal of pain during all rehearsals and performances for my last year. The vibrations of the big horn really make me miserable, even though I love playing jazz on the bari sax. I am still playing bari sax as much as I can in the Grand Rapids Jazz Orchestra, but playing alot more flute, alto and tenor sax now. At least flute does not bother me much, and I am getting very serious about it. For some reason, the act of fast tounging really bothers me on any instrument. My trigger points seem to change from left side of my mouth/lip to toungue, to teeth, to jaw. I may also have some tmj.