For over month I've had dry throat, constant mucus and mild pain sometimes when swallowing. No strep, no fever, no tonsils. One dark vein in throat

Throat. Sounds like you have been evaluated and you do not have strep. Other possibilities could be allergies, mycoplasma infection or
simply dry mouth. Dry mouth can be from aging - simply do not make as much saliva, medication- anticholinergic effects or conditions-Sjogrens syndrome for example. .