I would like to add,
The problem l have now is very similar to what you described. Whenever I tried raising my voice or exhaling more forcibly, everything would seal off. You said this was diagnosed as valopharyngeal insufficiency. As I understand it, valopharyngeal insufficiency is when the valve can't close properly because of a cleft palate letting air out the nose when it should not.
In my case the opposite effect happens. My endoscopy showed the valopharyngeal valve closing while I'm trying to let air out. Basically it closes when it shouldn't. My doctor said this was a functional issue: I was the one commanding these muscle movements because I had learned to breathe wrong.
However when I started using the stents and my nasal passages were forcibly opened, I could speak comfortably and the valopharyngeal valve stopped closing unnaturally. This suggests that the problem is not functional but rather mechanical. My best guess is that trapped air builds up in the narrowing nasal passages potentially creating an area of high pressure beyond the valve. Air flows from high to low pressure so when the outgoing air influx from the lungs stops or reduces the only thing that air could do is reverse direction. The valve could be closing as an autonomic reflex to stop that reverse flow. When the air beyond the valve slowly escapes through the nasal passages and pressure decreases, it would then open again which would also explain the short bursts of relief. I've yet to see another physician and this all just speculation.
You also spoke of over-exerting the muscles for speech and how that causes headaches. Did ever feel unnatural vibrations in your nose when talking? I feel that the energy I input for speech is not proportional to the output volume and I 'm guessing a part of that sound is getting trapped inside the nose because of the congestion and lost as vibration.

Thanks for responding and hope you feel better.