Tonsil crap....
I don't know if this may be your problem but research the net as I have done.
Symptoms. As you describe. Please examine your tonsils and if they are large and "puffy" and appear to have holes in them, these "white spots" maybe the visible end of plugs of waste matter, stuck in the holes or fissures of your tonsils and are made up of food debris, oral secretions and bacteria, and not very nice bacteria, either. These plugs are far from nice, they can (and do) smell diabolical, and (with a certain amount of practice where you suppress your gag reflex) can be prised out with your finger nail or other suitable (no, not that one) instrument. I am being serious now!!
I have carried out (as a layman) considerable research into this well known affliction, because I am so afflicted. .and have been so for many, many years, as is most likely 90% of the people with halitosis.
Please look up tonsilloliths on the net, and there will be the answer, I believe, to your query.
What do you do about it? Well, having your head cut off is one, which is not advised, even a do-it-yourself job!
Also do not depend on any face covering as it does not disguise the smell! But please, in the interest of those you converse with, do pursue a rigourous dental and oral hygeine, give your tonsils a good poke now and then, and above all, get it seen to. ASAP. I can smell you from here!