Prior to the start of the ARW, there were numerous acts of minor terrorism by various rellion group, most notably the Sons of Liberty led by Samuel Adams and others (remember the Boston Tea Party). Once the organized war effort fails, and depending upon how the British deal with the American leadership - signers of the Declaration of Independence and the upper level officers of the Continuental Army (starting with Washington), I think there will be an almost constant underground war conducted mostly with acts of more and more violent terrorism.
If the British decided to make an example of the Americans and deal with them very harshly (leadership rounded up and imprisoned and/or executed, thousands of British troops occupying the colonies "to keep the peace," and the removal of most rights of freedom that the colonists had enjoyed prior to the revolution), then you would end up with a very different America. Violence and acts terrorism would escalate as colonists did what they could to force the British out. Colonial governors and other politicos appointed by the British crown and Parliment would be in constant danger, and in many cases would be killed or kidnapped. Such positions would become very undesirable by loyalists and English born citizens. Immigration to the colonies would almost completely dry up as they would not be seen as a place of hope and freedom, and in fact, thousands who could afford to would emmigrate back to England or Scotland. Because of the acts of terrorism, the English grip on the colonies would become even tighter. If this situation lasted for 12-15 years, then eventually, the American colonies become secondary to British Canada and are eventually linked with Canada. Louisiana either stays with the French or is really given/sold to Spain. The North American continent looks nothing like it does in OTL.