Hello,
From your description, the most likely explanation is that what you are feeling is a torn, partially detached
hymen.
It doesn't have sensation when you pinch it but you feel it when its pulled because the nerves that you are feeling are inside where it's still partially attached, not in the piece of tissue that's hanging.
This situation of a hanging remnant of hymenal tissue has happened to many women.
Now what to do about it. In some cases the rest of the hymen (where it is attached at the "hymenal ring") will detach after further vaginal activity (tampons,
masturbation, intercourse), but often it has to be removed by a gynecologist. It is a fairly quick, simple procedure for the gynecologist to remove it.
My suggestion would be to see a gynecologist to have it removed. These doctors are used to this problem so there is no need to feel embarrassed telling them about it. But I know that's easier said than how you may be feeling right now!
My best wishes to you - take care.