How hard is my air soft shotgun hitting?

Well it's only 1 N of force but because it's applied over a small area, it does more. Like I said, my answer is an underestimate because I took the area based upon the diameter (which is the largest of the BB's cross-sectional areas) and as the area gets smaller and smaller, the force per unit area gets closer and closer to infinity.

The force applied is independent of the area it is applied over.

Your equation isn't considering drag. Also, your acceleration assumption is wrong. I don't think it's possible to determine acceleration because the force applied is based on the acceleration change between the moment it impacts the skin and the moment its velocity is zero, not the moment between the time it exits the barrel and then stops.