The authorization by Congress to use all necessary military force constitutes the functional equivalent of a formal Declaration of War.
The USA has formally declared war a grand total of 5 times under the Constitution, 4 of those after hostilities had started.
We have engaged in military action literally hundreds of times, with tacit or explicit congressional approval in most of those cases. The very first example was the Quasi-War with France, where President Adams fought without a formal declaration of war. This allowed both sides to pretend they weren't really at war.
As far as using military force against American citizens without a Declaration of War, this goes all the way back to Washington and the Whiskey Rebellion.
As I've said elsewhere, the traditional idea of a formally declared war is that it can be declared only against another nation-state. We need to either expand the concept and formally declare war on al-Quaeda, which would be fine by me, or recognize that in a time when our most aggressive enemies aren't nation-states, Declarations of War are obsolete.
Do I trust Obama to use such powers wisely? Hell, no! But that doesn't mean the powers are wrong, only that we elected a bad user twice.
Back in the day, Congress used to issue Letter's of Marque and Reprisal to private persons. These letters allowed Americans to attack the forces of foreign states or even non-state actors such as pirates for a bounty to be paid by the US government. These letters basically commissioned privateers to go attack hostile groups or individuals. I don't know if they could be used against US Citizens. Letters of Marque haven't been used for about 150 years, and are thus considered "obsolete" by some other countries, but in theory they could be revived by Congress to deal with non-state threats outside of the US.