Dear Zambia,
The correct phrase is without further ado.
An ado is a hubbub, a bustle, a flurry, or a fuss. Another common phrase, from the title of a Shakespeare play, is “much ado about nothing.”
“Adieu” is the French word for “goodbye.” English just borrowed it directly from French.
Ado” was originally a contraction of the words “at do,” which was another way of saying “to do” because some of the languages spoken by the Norse invaders in northern England used the word “at” the way we use the word “to.” According to the Oxford English Dictionary, it looks like “ado” is still used to mean “to do” in Scottish English and maybe in northern England. Here’s an example sentence from a Scottish Dictionary published in the 1970s.
Substituting "adieu" for "ado" is what linguists call an eggcorn, confusing two words that sound the same, especially when the substitution makes a bit of logical sense.