The first option ("The war for independence officially over, the American Congress in 1784 turned its attention to the business of building a new nation.") is definitely better. The second sentence ("Now that the war for independence is officially over, the American Congress in 1784 turned its attention to the business of building a new nation.") has several errors:
- now and 1784 shouldn't be used together, since it's (currently 2012)
- now that indicates that something is complete. The information that follows in the main clause should be in a verb tense that is not "complete" (i.e., past tense), meaning we could use either a present tense or a future tense. For example: "Now that the war for the next President is officially over, the American people will be able to turn its attention to the business of continuing to recover from the Great Recession of the 2000s.
Hope that helps.
Erin