We do not know whether serving as president after 1797 would shorten or lengthen Washington's life, but it is unlikely he will serve out his full twenty-year term. Depending on just when he dies, he is presumably succeeded by twenty years of either John Adams or Thomas Jefferson. If Adams is elected in 1798-9 at the height of the anti-French hysteria, his election will be the product of a passing mood soon violently reversed, and he will soon be frustrated by a hostile Republican Congress--yet the country will be stuck with him for twenty years unless he steps down or the Constitution is amended...
Even if the Constitution if amended, as sends likely, it is likely amended for
future Presidents, specifically excluding the current office holder. Just as the 22nd Amendment excluded Truman from the term limit.
The other point you raised, that Adams would be elected based on anti-French hysteria. I don't think that would happen in a world with 20 year terms. The electorate as a whole would, I think, place much more weight on having a steady, moderate hand on the tiller. With no way to vote out someone you don't like for 20 years, elections likely are given much more thought and consideration than they are today. We would never see someone like Trump even be considered for the office ITTL. Likewise, people like McCain, HRC, etc would all be considered far too old and highly unlikely to finish out their term.