I had this experience daily in the hospital. The first night after transferring out of the ICU, it got so bad, it turned into real vomiting. Even now, over a year after the surgery, if I am not quick enough to cough/spit up the mucous, I can feel the beggining of the gag reflex. All I can say is that it does get better with time. Also learning to sip water/swish it out in a forceful spit helped. If Gordon can gargle, that's a big help.
finally, an index finger wrapped in a tissue swabbed around the mouth is my old reliable, physically removing the sticky mucous.
Good luck.
Charm