I had a similar experience, only mine was worse, I even got a couple abscesses. Do you have any concept how horrible that is? So, I'll start off saying yes, you should go to the dentist. Yeah, I didn't either.
Here's what I did do; be forewarned this is going to take dedication and effort on your part but you'll be happy you did it in the end.
First of all, if you use any kind of teeth whitener, mouthwash or solution, stop immediately. Anything that contains peroxide will eventually make your teeth sensitive. Fortunately the sensitivity will reverse once you stop using it.
Second: Start flossing your teeth when you brush them. Every. Friggin'. Day. Twice even.
The pain in your teeth could be an abscess. If the pain is there when no cold or sweet is touching the teeth, it is not merely sensitivity. Believe it or not the absolute best way to relieve an abscess (short of going to the dentist) is to put a regular good old black tea (think Tetley or Lipton) teabag on it for at least an hour. I'm not sure if it's actually the tea (the tannin maybe?), the teabag itself, or a combination of both, but this will draw out what is causing the pressure on the nerves in your teeth and heal the abscess. You may have to do it more than once, but often once will do it.
Last but not least, gargle with the most effective, and believe it or not cheapest, mouthwash of all... straight vodka (you obviously want to buy a large bottle of cheap vodka for this, not Stoli's or Finlandia). This is pretty much like gargling with Listerine without all the additives, since what makes it work is the alcohol, and it is infinitely more effective.
Stick with this process for as long as it takes, and never stop flossing for the rest of your life. You'll be good to go and at least be able to put off going to the dreaded dentist for this particular problem. My teeth are in great shape now, but it was work to get here!