Your symptoms sound a lot like mine. I had mono almost three years ago and had a terrible cough with it. Now, it seems especially when I am worn out, not getting enough sleep or whatever, I cough. Usually when my cough is bad, I am more fatigued than usual (or maybe vise versa). I have not been to the dr in a while, as I have been feeling better and the coughing spells seems to be fewer, so I have no idea what my EBV numbers are right now. I did go to a dr over a year ago for lung testing - for asthma in particular, but all test were negative and I was told my lungs were fine.
I wonder if the cough, drip, sneezing is indeed related to EBV. I have wondered too, if my allergies to certain things like mold are worse now - maybe unrelated to EBV... The roof leaks in the building where I work and I know that can't be good. I have had fussy sinuses for a while, but I do think it is all worse since having mono. I don't think you are contagious, probably a coincidence that your co-workers seemed sick while you were there. Also, the power of suggestion can be very strong.
Good luck with finding out what the root of your cough is. If you get a diagnosis, post it here; I would be interested.