The time that you would have had to react to the news that France had sold Louisiana to the United States, may not have been months, but probably weeks or days. In the early 1800's news traveled slow by today's standards, and the news would have been slowed down even more due to secret treaties and colonial revolutions in Haiti. It helps to start the answer to your question in 1800, when Spain agreed to transfer control of Louisiana to France, as Spanish officials end up playing a role in the transfer of power and the answer to your question.
The Louisiana Territory was controlled by Spain in 1800 and ceded to France by the secret Third Treaty of San Ildefonso on October 1, 1800. The terms of the treaty were later affirmed publicly by the Treaty of Aranjuez on 21 March 1801. However, the terms of these treaty did not go into effect until later. In fact, Juan Manuel de Salcedo, the last Spanish governor in Spain served until November 30, 1803.
So, what was life like in New Orleans during this lengthy transfer? Well, Pierre-Clement de Laussat the French official who was supposed to prepare for the transfer of power from Spain to France would have arrived in early 1803 before the sale of Louisiana to the United States. A a french government official and autobiographer, he has left a lot of information that proves useful when trying to understand what New Orleans was like in 1803.
Laussat was sent to prepare the colony to become the supply center for troops that would be stationed across France's Caribbean empire. While the sugar plantations and an economy fueled by people in slavery were the center of Napoleon's ambitions, Louisiana would be the hinterlands that supplied the island. There were also dreams of controlling the North American fur trade through Louisiana. However, Napoleon proved unable to quell the revolutions in Haiti, and gave up on his colonial ambitions. This is what led to him selling Louisiana.
During this time, Laussat would be making connections in the government and trying to win over Spanish official and powerful residents. There were incessant rumors that Spain would refuse to turn over Louisiana because France had not upheld it's end of the bargain. In fact, the aforementioned Governor Salcedo was one of the many pushing for Spain to not give up Louisiana. At the same time, there were also rumors that the Americans may come down the Mississippi River and take New Orleans by force. The year prior in 1802, another Spanish official had shut the Port of New Orleans to Americans. While the latter issue was resolved, Laussat continued to struggle with the rumors of Spanish refusal even after the new of the Louisiana Purchase reached New Orleans.
So what of the Louisiana Purchase? Well as Laussat worked to ensure that Spanish officials would hand over Louisiana once the powers in Europe finalized the details, he received shocking rumors that his work would be almost for naught. Records show that he wrote to France to inquire if the rumors were true in late July and that he did not formally receive word confirming the sale until August. The on-going legal struggles between France and Spain muddled any sure sense that France or the United States would actually come to possess Louisiana. In fact, it was in October that the French ambassador to the United States wrote Laussat to assure him that he would receive orders to take possession of Louisiana from Spain on behalf of France (up to this time, Laussat was only preparing the situation for the arrival of a general who has the orders to take charge on behalf of France). The ambassador also provided guidelines on what to do if the Spanish refused to transfer power. The uncertainty was furthered by the fact that some Americans argued that Thomas Jefferson could not buy land without congressional approval. With all this uncertainty and arguments, it's easy to see how Laussat could have first doubted the news as would have many others.
So in a swirl of rumors and political intrigue, what did life look like for the average French (or more likely Spanish) citizen living in New Orleans? Life and government continued to be administered as usual. Like many frontier towns and port cities, there was a hustle and bustle of immigrants and traders. During all of this, Laussat kept a diary and on one day in May he wrote,
“The products of Louisiana are already quite considerable. Wherever the Anglo-Americans settle, land is fertilized and progress is rapid. There is always a group of them who act as trailblazers, going some fifty leagues in the American wilderness ahead of the settlers. They are the first to migrate to a new area. They clear, populate it, and then push on again and again without any purpose other than to open the way for new settlers. Those who thus forge ahead into unknown places are called backwoodsmen. They set up their temporary shanties, fell and burn trees, kill the Indians or are killed by them, and disappear from this land either by death or by soon relinquishing to a more stable farmer the land which they had begun to clear. When a score or so of such new colonists have congregated into one location, two printers arrive—one a federalist, the other an anti federalist—then the doctors, then the lawyers and then the fortune seekers. They drink toasts, nominate a speaker, set up a town and raise many children. Finally, they advertise the sale of vast tracts of land, attracting and deceiving as many land buyers as possible. They exaggerate the population figures so that they quickly reach the sixty thousand souls entitled to form an independent state and be represented in Congress. And so another star appears on the flag of the United States! A district under the Spanish or French regime might begin, end, start again, and get lost again, and so successively until its fate is sealed—permanent existence or annihilation. Under the Anglo-Americans, a newly born state may thrive with more or less prosperity, but it will never decline; it keeps on growing and strengthening. One can hardly realize that forty years ago, on these vast expanses of land from the shores of the Mississippi to the Alleghenies, there was not a single farmhand to cultivate the soil. Today, these same regions flood the New Orleans market, by way of the Mississippi, with their abundant harvests.”
It would seem that life in New Orleans was marked by the arrival and departure of American colonist heading up river to the interior. The Spanish government continued to explore, maintain, and exploit the frontier. After the transfer of power Laussaut used his 21 days as governor to appoint the first municipal government for the city of New Orleans, establish a city fire department, formulate police regulations, and begin the preservation of public archives. The sources vary on how people actually felt during this transition time, but they point to the fact that feelings varied from surprise to anger to apathy. The majority French population did face skepticism from the Americans. Only a few decades earlier in 1766 there was a major Creole rebellion against the Spanish rulers. The records also show skirmishes between French and American residents before and after the transfer.
Finally, an interesting tidbit, while there was a period of French governance, in St. Louis the administrative seat for Upper Louisiana, power transition from Spain to France to the United States in the same day. March 9-10, 1804 is know as Three Flags Day. On March 9, a flag raising ceremony transferred power from Spain to France, on the next day a flag raising ceremony transferred power from France to the United States.
Sources:
LOUISIANA: EUROPEAN EXPLORATIONS AND THE LOUISIANA PURCHASE A SPECIAL PRESENTATION FROM THE GEOGRAPHY AND MAP DIVISION OF THE LIBRARY OF CONGRESS
Louisiana, Napoleon and the United States: An Autobiography of Pierre-Clément de Laussat, 1756-1835 (referenced here on google, but cited in several other sources)
Citizen Laussat: A Retrospective on the Louisiana Purchas
Jean-Marc Olivier. Bernadotte, Bonaparte, and Louisiana: the last dream of a French Empire inNorth America. The Impact of Napoleonic Empire in the Atlantic World, 2009, France. pp.141-150. hal-00974220 (link)
Lastly my apologies that this response has run on so long and come so late. As a native New Orleanian I had inchoate inkling of the answer based on stories I've heard and have been told, but the rabbit hole of research quickly dispelled a lot of my notions.
What type of governing body was in Louisiana? Were they aware/part of the negotiations?