What is the difference between turgid, tumid, and tumescent?

Tumid and turgid appear to be very similar in meaning both etymologically and in modern usage, but to my ears and by association with tumor, tumid carries a more negative, serious connotation, while turgid is a swelling but not necessarily diseased in itself. As @Nate notes in the comments, turgid is used positively in biology. (These are my impressions. Take them cum grano salis.)

Etymologically, tumid meant "morbidly swollen", i.e., swollen because of disease, and is cognate to tumor. Turgid also meant "to swell" and originated in medical contexts. The Latin roots of these words (tumere, turgere) also marks them as being very close in meaning if not synonymous in many contexts, with tumere appearing to be more common in classical Latin.

Tumescent is also cognate with tumid. The "sc" in the middle of Latinate words typically indicates a changing state, e.g., fluorescent, condescend, convalescing, adolescent, effervescent, emasculate, etc. (Note this doesn't apply to prefixes that make the letter combination by happenstance, e.g., discombobulate, disconnect, misconstrue, resuscitation, etc.; to compound words where the "sc" was at the beginning of one of the pieces, e.g., kaleidoscope, subscribe; or to words from Greek or other languages, e.g., proboscis.) Here it makes the meaning progressive rather than tumid's static sense: swelling rather than swelled.

Tumescence connotes being filled with water in a good way as in your example the penis fills with blood becomes tumescent and achieves an erection which is its function.

Turgor can also be a good thing it is what keeps plants erect. Without enough water plants would wilt. With turgor they stand erect and grow towards the Sun.

Tumid also means swollen but is always pathologic. For example bloated dead fish would be tumid. A distended abdomen of appendicitis would be tumid. This patient is at risk of death.

So if I were using these terms that were originally medical and scientific and wanting to apply them to someone's manner of personal expression or writing I would do this:

I would use tumescent to describe a temporary overflow of pride that someone uses to bluff their way through a temporary situation. Like athletes before a game.

I would use turgor to describe someone who is always overdoing it, but not necessarily in an evil way. For example Muhammad Ali bragging all the time about what a great boxer he is. Sure it was prideful but it worked for him and it didn't hurt anybody.

I would use tumid to describe an over expression that has become pathologic and will soon result in the demise of the person or civilization. For example the fall of the Roman Empire. Or the idiocy of a dictator eg Hitler.