Audra, if you want a good fiction novel on the British Navy during the classic age of sail, you’ll be hard pressed to find better than Horatio Hornblower by CS Forester. Especially as it’s been quoted as an influence on both Gene Roddenberry and Shatner when creating Star Trek and also a big influence on Nick Meyer when he was doing Star Trek 2, Wrath of Khan. It was also David Weber’s inspiration for Honour Harrington and in one of his books she’s actually reading a Hornblower story.
They are good swashbuckling stories and reasonable portrayals of how the Navy operated, although the hero is not always the most sympathetic of characters. Start with The Young Hornblower and go from there.
However, I don’t think they particularly support the claimed trope of douche admirals. A number are featured of varying competence and niceness, but I don’t recall any that were particularly douche like and some are great supporters of the hero in fact. I’d say the two biggest douchebags in the series were a Captain and a midshipman.
As someone else has pointed out, in the Napoleonic era the Navy was pretty much a meritocracy and unlike the Army there wasn’t much in the way of people buying their way to power. There had been in earlier times, but after a few embaressing disasters the Admiralty did extensive reforms to make the Navy more professional, which is one of the prime reasons the Royal Navy ruled the world’s oceans in that period.