What effect did "common sense" have in the colonies?

Here's some extra information you may need.

First, you are right that the Continental Army came from all of the colonies, the idea is that this represented the continent of (what was then known as) the new world. The idea is that only colonized areas are part of this continent.
Next, the battles of Lexington and Concord were incredibly important for a number of reasons. For one, they solidified the threat of the colonists' army, as (although small) it was able to take on the most powerful military in the world at that time. These battles also included the first real formal fighting between the British soldiers (regulars) and the colonists, this represented the point of no return and to most all colonists, war was inevitable at this point.
Thirdly, the battle of bunker hill resulted in a win for the regulars, but also in mass casualties, as death and injury on their side was in the thousands, and the colonists only lost some 100 men. The colonists were hugely unprepared for this battle, as they had very little ammunition, and although fighting valiantly, were forced into retreat after fighting became unsustainable.
Fourthly, Patrick Henry sided heavily with the colonists, and he rallied soldiers, screaming: "Give me liberty or give me death!"
This next one is unclear to me, as views and ethics changed from person to person. If tough could explain this one I would be happy to help, and if not there may be others to help.
Anyway, then the Declaration of Independence was written, yes to show separation from the British empire, but really to get support from the other empires of the time. It is important to remember that, although the largest, the British empire was not the only empire, and the French empire especially was angry with the British after being expelled from the 'new world' as a result of the French and Indian war. So why not get their help? The colonists needed more supplies, better plans, more troops, and needed a place to trade with after the war, in order to sustain their economy. The Declaration of Independence was a call to arms for the French empire.
Jefferson believed King George III to be unfit to rule due to grievances against the colonists. Colonists were tried in courts far from home with biased juries, taxed harshly on nearly ALL goods, and more than that, they were not represented in parliament. The grievances on the colonists were more the fault of British parliament than anything else, but King George III was more harshly blamed than anything else.
I should preface this, I know of one reason, however, the argument could be made that this reason is made of other reasons. During the signing of the Declaration of Independence, all members of the Continental Congress must approve of the document for it to pass, but when the document was brought up, many Southern states, whose economies relied on slave labor and agriculture, struck the bill down, forcing Jefferson to rewrite the declaration, removing the part about slavery to appease those continental congressman.
Natural rights, such as life, liberty, and the pursuit of happiness, are "inalienable" and thus cannot be sacrificed or given away to others. It should be noted that, like all other things of the time, this only related to Caucasian men, but was later expanded to represent women and all other ethnic groups. 
Contrary to common understanding, the Declaration of Independence was signed August 2nd, 1776, almost a full month after many believe it was signed. 
The Olive branch petition was an attempt for colonists to make themselves respected and treated as subjects under the British crown. Many patriots even wanted to be respected under British authority, very few people wanted freedom from Britain, they did, however, want freedom from tyranny. 
The colonies came together in the second continental Congress (the first one was a failure) and unified for their rights.
The Second Continental Congress established the Continental Army, a group of civilian soldiers lead by George Washington to fight back against the regulars.
Finally, to Great Britain/the British empire, the signing of the Declaration of Independence was an act of high treason/treason against the king (either are acceptable.)

I hope I helped you! (Yes, I know it is a lot)