I had never heard of Sensodyne causing tooth sensitivity before receiving your question.
Is it your teeth or your gums that hurt? If it's your gums, you may be reacting to something in the toothpaste....sort of like an allergy.
I get many good reviews from people who use Sensodyne. They say it really helps make their teeth less sensitive.
But, it seems there are others who have had a bad reaction to Sensodyne and other toothpastes that are made to improve tooth sensitivity. You can read their comments here. These mixed reviews don't mean Sensodyne is a bad product, or that there is a conspiracy in the dental profession to make your teeth hurt so you will have to see your dentist.
It just reinforces the fact that no one products works for everyone. It's best to choose one that is made for your particular issue and avoid ingredients that you may be sensitive to.
It is possible to be allergic or sensitive to anything. You can read more about toothpaste allergies here.
Sensitivity to certain toothpaste ingredients is not unusual. It can cause gum soreness, sometimes it even causes gum tissue sloughing (peeling) like you peel after a sunburn. You should stop using it and see if the problem goes away. If you don't have sensitive teeth you don't need it anyway.
I like Prevident 5000 for tooth sensitivity and cavity prevention. Prevident has more fluoride than over the counter toothpaste and not only improved sensitivity but makes teeth more resistant to cavities.
Thank you for writing and good luck,