The United States pretty much did go to war with France during John Adams' administration.

Although it is often called, "The Quasi War", it was authorized by Congress, and involved battles at sea and the capture of American and French ships. It is perhaps best looked at as an American declaration of limited war (limited to war at sea).

After the French Revolution, the US stopped paying its debts to France, (arguing that it owed money to a regime which no longer existed). This led to the French seizing American merchant ships.

This led to the US building a navy to protect her shipping, and Congress passing (on July 9, 1798) "An Act further to protect the commerce of the United States".

This act was effectively a declaration of a war at sea. It authorized the President to ,"instruct the commanders of the public armed vessels which are, or which shall be, employed in the service of the United States, to subdue, seize, and take any armed French vessel which shall be found within the jurisdictional limits of the United States, or elsewhere on the high seas."

http://scholarship.law.edu/cgi/viewcontent.cgi?article=3342&context=lawreview

It also authorized the President to grant commissions to privateers to attack and seize French shipping.

It also authorized the taking prisoner of crews and passengers on French ships, and transporting them to the United States where they would become the responsibility of the US government.

The US navy then went to war. The frigate Constellation defeated and captured the French frigate L'Insurgente, other US warships and privateers captured many French privateers and merchant ships.

The end of this naval war came as Napoleon became leader of France. He decided he did not want the complications of a naval war with the US. The "Convention of 1800" was the treaty between the two countries which ended the "Quasi War".