How can I cure my bad breath from nasal drip?

So I have a problem. I've developed really bad breath that i've been ignoring now for some time. I recently got dental insurance and went and had a cleaning, xrays, and 6 cavities filled. The dentist says theres nothing else that needs to be done and they'll see me next cleaning.

I didn't ask the dentist about the issue because she was busy and didn't seem very open to questions. I also have a feeling that this is more of a doctor/ENT type problem as it doesn't have much to do with my teeth.

Basically when I swallow I can feel small amounts of stuff coming down from the nasopharynx area and causing foul breath. My snot is very sticky and so this drippage hangs out right on the border of the nasopharynx/oropharynx and sits there smelling up the back of my throat. It comes and goes, sometimes not present at all, but sometimes so disruptive as to make it almost feel uncomfortable/choking/gag. It is extremely hard to dislodge, clearing my throat does nothing, swallowing extremely dramatically seems to help sometimes.

I am desparate to get rid of this problem. Any advice would be very welcome. I wish there was a way to blast water upwards from inside my throat and blast all of that gunk out of there through my nose. Does such a thing exist?

This picture shows a diagram: http://headandneckcancerguide.org/wp-content/uploads/2013/02/04a_throat_overview-CROP.jpg