Tonsil stones. Did you have strep throat a lot as a child? This is a lasting affect of that.

Sounds like you might have tonsillitis!

It's a tonsil stone. Not really a stone but a collection of bacterial debris that's collected on your tonsils and then comes loose, and often to come out of your mouth. Hence the disgusting smell. They are relatively common. Gargling regularly with a medicated mouthwash can help stop them forming.

tonsil stone

if you leave it sit on the counter for awhile, it will turn brown and hard

try to see how big a one you can cough out

Could be parasites.

Did it have a head?