So sorry you are feeling anxious. I wouldn't worry about a shallow cough too much unless it lasts a really long time, but I'm no doctor! If you are worried enough you should go to a real doctor.
It's hard to get over sinus infections. Since the sinuses have very little blood supply not much medication gets to them. I have pretty much chronic sinus problems even after sinus surgery, so it's not worth going that route, IMO.
The most effective thing for me is to do daily sinus wash, at least in the morning, sometimes in the evening too. Put "sinus wash" in the search box at the top of the page for tons of threads. I like the NeilMed bottle, but my sinus drain passages were opened up and I can wash them out pretty well. If you haven't had the surgery, which I don't recommend, you may just end up rinsing out the nose area. But even that may be helpful.
Remember that the maxillary sinuses drain upward, so after rinsing I make sure to lean over for a good while or get embarrassed when I bend over and get a gush of water out in public.
The most important thing about sinus washing is to use sterilized water. The brain eating ameba has been found in public water supplies even in the northern states. I just get a gallon of distilled water. "Spring water" isn't necessarily sterile. Use the right amount of salt and baking soda for the size bottle you use.
My sinuses get running first thing in the morning. I don't think talking on the phone would be relevant. Mine dry out at night too. A quick rinse in the evening sometimes might be helpful. If you get "crusties" you might try a dab of AYR around the nostrils. It's OTC. Don't use petroleum jelly.