I feel for you, it's an awful condition. I don't normally get sick but a few times a year I have experienced persistent coughing VERY SIMILAR to yours -- except I usually lose my voice too. No meds have been effective to cure it (even mucinex and a prescription-grade cough syrup). For me, without fail, what CURES a brutal cough caused by persistent sinus drainage is:
SINUS RINSING aka SALINE RINSING aka NASAL RINSING.
It's not fun, but it doesn't hurt and (for me) it's been 100% effective. I learned about it from Family Practice Assoc at Avista Hospital, you can search the web and find plenty of info on this, but it's very simple to do. You can make your own warm saltwater solution at home and get a snubbed nose bulb (from Walgreen's). Frankly it's easier to just buy a bottle of saline nose spray at any grocery or drugstore.
1. First I usually gargle warm salty water (add enough table salt so that it tastes salty without burning). While gargling, vocalize Ahhh-Eeee-Ooooo to make it more effective. You can feel the saltwater dissolve the gunk in your throat. Spit and try to get the gunk out.
2. Next, squirt the saline up each side of your nose til you can feel it working (you'll know). If you really want maximum effectiveness, inhale when you squirt -- since I don't like how that feels, I usually don't. After each good spray, I usually blow my nose.
3. Repeat 2-5 times. Your goal is to have clear sinuses and a clear throat.
This procedure isn't glamourous, it isn't fun, but it is not painful -- although it feels very funny the first few times.
Do this 3-6 times a day until your cough is gone. If this hasn't cured your cough in 3 days (and you've actually been doing it 3-6 times a day) ask a doctor to check you for sinusitis.
Be well.
..J.
p.s. for me, rinsing is very effective during the day. But until you are cured, I recommend Tylenol Nighttime Cough & Sore Throat (blue liquid). Just remember to brush your teeth after taking it, since cough syrup can damage your enamel.