In the story, Ou-dis-sun does refer to the Hudson River, as the other educator has mentioned. In the story, Ou-dis-sun is also called the "Sacred, the Long." The narrator proclaims that Ou-dis-sun could "eat all the streams we know and still be thirsty." It is indeed the longest river in New York state.
Interestingly, by the time Henry Hudson discovered the Hudson River Valley in the 17th century, the area was already home to the Mohicans (Mahicans) and the Lenape or Delaware Indians. The Mohicans called the Hudson River Mahicannituck, meaning "great waters that are never still." The Hudson River was important to both Native American tribes. The river provided plenty of sturgeon, bass, and herring for the people, and the river beds yielded oysters.
According to the narrator, John, in Waters of Babylon, no one from his tribe has ever seen Ou-dis-sun. He considers it a magical place. Interestingly, he describes the river perfectly; at one time, the waters were indeed of great importance to the "gods" in the Dead Places.
As for the Great Burning, this refers to some past, cataclysmic Armageddon, when "gods" warred with "gods." The Great Burning is a metaphor for a nuclear holocaust, which resulted in the complete eradication of human civilization in the Dead Places. John implies that the nuclear war rendered the Dead Places devoid of its "gods." He mentions the "mist that poisoned" and how that poison still lingers in the ground. The narrator's second sight allowed him to see how the Dead Places were destroyed by this poisonous mist (likely radiation from nuclear bombs). Basically, John witnessed the destruction of humankind through a nuclear holocaust.
As for ASHING, this likely refers to Washington, the first President of the United States. In the story, John states that the statue bearing the words ASHING had been shattered into pieces. Interestingly, John also found a cut stone with the letters UBTREAS nearby. In the 1700s, the City Hall of New York became Federal Hall; this was where George Washington was inaugurated and where the Bill of Rights was passed. Federal Hall was demolished in 1812. In 1842, Federal Hall was rebuilt as a United States Custom House and later served as a United States Sub-Treasury from 1842-1920.
Federal Hall was built in the Greek Parthenon style. This is why John sees a "great heap of broken stones and columns" in one area. The Parthenon style of architecture favored columns. A statue of George Washington stands in the front of Federal Hall. Based on these facts, we now understand why stones inscribed with the words UBTREAS and ASHING were found in close proximity.
Well, it helps if you know ahead of time that the Great Dead Place, where John goes on his journey, is actually what's left of post-war New York City. This can be inferred from some of the things you've mentioned. The first clue is ou-dis-on. This is actually a distorted version of Hudson, as in the Hudson River, which runs through the heart of NYC.
The Great Burning likely refers to an explosion caused by some sort of bomb during a great war in the past. Additionally, ASHING is likely part of the name Washington, which John sees when looking at a statue of the famous first president.
It's really a fun story when you know what you're looking for. Best of luck with the rest of it. Just remember that even though John and his family seem very primative, they're actually in the future, far down the road from our present time.