In 1797, were relations good between France and the United States because Britain was their common enemy?
In short, no.
While the French and Americans have generally enjoyed good relations, that particular year, they did not.
In the earliest days of the republic, the US had two political parties: the Federalists and the Democratic Republicans. The Federalists favored relations with England over France (even during the Revolutionary War, their long-term plans involved extensive trade ties with Britain) and the Democratic Republicans were just the opposite; their standard bearer, Thomas Jefferson, was a Francophile and had been America's ambassador to that country during the earliest days of the French Revolution.
The Federalists, as powerful and influential as they were at the nation's founding, only elected one president, John Adams, and he served only one term. (George Washington was more in line with their beliefs than with the other party's, but was not an actual member of the party.) Adams was particularly hostile to the Jacobins, the perpetrators of the French Revolution, and voided agreements with France on the grounds that they had been made with the government of Louis XVI and were nullified by his execution. Adams' single term began in 1797, the year in question.
Jefferson, who fancied himself to be the inspiration for the French Revolution, had warmer relations with the French--but by the time he took office in 1801, the Revolution had fallen apart and Napoleon had seized power.