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It's easy at least to show that part of your premise is incorrect: so far as I know, the North Koreans and South Koreans have never made any agreement to any partition of what they each regard as their entire country, except for a provisional military armistice line. The 38th parallel line was decided in August 1945, by two junior American officers working from a National Geographic map. For some reason, it was immediately agreed to by the Soviet Union. Not only were no Koreans consulted -- the Provisional Government was entirely ignored then and I believe afterwards -- no other allies were consulted, and nobody with knowledge of Korea was consulted. A joint Soviet-American commission was formed, but it deadlocked fast. In 1948, the U. N. General Assembly declared the Republic of Korea (South Korea) to be the legitimate government of all Korea. Low-level guerrila warfare and incursions continued until the North started open war in 1950. A significant factor for the course of the Korean War was Rhee and Kim trying to crush the other's system. There has been warming and cooling since, and armed incursions and actions since (pretty much by the North), and agreements on basic principles of reunification, but that hasn't gone far.

The Geneva Agreements of 1954 did actually somewhat “split” Vietnam. Since the French involvement in Indochina had began to become extremely unpopular at home, the new French prime minister (Pierre Mendès France) promised that he would gain a ceasefire in Indochina within one month of his rise to power. Lighting a fire underneath his own feet, Mendès France began have secret negotiations with both the anti-French Viet Minh and Chinese.

Eventually, this led to Vietnam being split along the Seventeenth Parallel, with a three mile DMZ between the two borders. This partition was accepted by the French, North Vietnam, UK, and China but was rejected by the South Vietnamese delegation. Still, the Geneva Agreements promised that elections would be held in 1956 (two years afterwards) in order to create a unified Vietnamese government.

Sounds like a pretty fair situation, right? Well, the kicker was that the South Vietnamese knew that North Vietnam had a much larger population, and thus the Communist government of the North would almost undoubtedly win the elections. Combine reluctance of the South Vietnamese with widespread corruption, political intimidation, and electoral fraud along both sides of the 17th Parallel and you get the ICC and UN stating “fair, unbiased elections were not possible”.

Back in the US, JFK gets elected in 1960, inheriting a generally lackluster American performance in the Cold War. The US loses the Bay of Pigs invasion and the Soviets are building the Berlin Wall brick-by-brick. JFK (and later LBJ), needing to preserve the reputation of Cold Warrior America, is determined to stop the spread of communism around the world.

This eventually leads to escalations on the military front, increased guerilla skirmishes in the South, brutal pogroms and political violence in both the North and South, and the Americans inheriting the nasty situation that the French were only too happy to hand over.