This is a long story so I will try to make it as short as possible.  When I was just 18 my cousin purchased a gun from a local gun store.  Soon after said purchase he needed some money and sold the gun to me.  Both of us knew nothing about gun laws or proper procedures.  I simply gave him the money and he gave me the gun.  This all transpired in New Jersey.  Now let’s fast forward to the present. (Approximately 12 years later)  I was cleaning this gun and it hit me like a ton of bricks that I never properly transferred this gun to my name.  The mind is a terrible thing.  I now live in AZ.  What do I have to do to properly transfer this gun to my name?  I am hopping that all I have to do is go to my local gun shop and fill out the paper work.  Is that correct?  Will my cousin have to fill out anything?

The Federal Government doesn't regulate private firearms transactions and you don't live in New Jersey anymore. There is no need to "transfer" anything. Quit worrying and go shooting.