The 1597 episode is during the 2nd half of the war, after the peace talks. The Japanese were far less successful in this part and failed to reach even the central part of the peninsula before retreating. The Ming were in the peninsula as well, unlike in the 1st half of the war, so it's doubtful the Japanese could've done much more than they did OTL, considering Hideyoshi is close to dead by this point and both sides were tiring of the pointless conflict that didn't see anyone gain a huge amount like they were promised. At best, the Japanese get a bit further into the peninsula but then it'd turn into a protracted stalemate like the end of the first half of the war. The war's not likely to go long past Hideyoshi's death; his megalomania was the cause of it and it's far more of a drain and slog than promised. The Jianzhou Jurchens and Tokugawa benefit from this (not fighting and getting time to consolidate), the Ming and Toyotomi suffer (spending resources fighting, aren't able to check their rebellious underlings due to lack of focus) and Joseon history manages to get that much more depressing with even worse famines. The greater devastation might prevent the Injo Coup by virtue of not being able to help either Ming nor Jin, however, which does help the Joseon a bit more by virtue of avoiding King Injo and the subsequent Manchu mistakes.
I think the relationship between the Korean Government and Korean populace would be heavily strained, especially after the execution of Yi Sun-Sin. My belief is that people will believe that Yi Sun-Sin was stabbed in the back by a corrupt, fractured government and a weak, cowardly king
I've heard some Koreans say precisely. One theory is that Admiral Yi, rather than die by being shot by a Japanese gun, actually offed himself following the Battle of Noryang Point because he figured it be better to go down as a war hero than to be executed for some nonsense the moment he arrived back in the capital (since King Seonjo tried to do that anyways). Not sure of the veracity of the claim but it's reasonable enough, seeing how Admiral Yi was treated
even when his service was needed by the court.
But yeah, overall, the second phase of the invasion's not likely to see the conquest of Korea, just based on how it petered out OTL. The first half, maybe, but it's still worth noting the Japanese went from Busan to Pyongyang in a few months and then lost everything to Hanseong in an equally short period of time. A decent army and strategy means no easy conquest and years of bogged down warfare for the invading force, with or without a brilliant leader.