If your character runs and then attacks in melee does the attacking character take a +4 to TN, and do his opponents then take a +2 to TN to hit them in melee?

I'd rule there's no modifiers. I don't remember if there are any rules for charges, but if there are they could be used.

We play that the Attacker gets movement penalties (e.g. +4 Running p. 108 or +1 Walking p. 108) but since the "Melee Modifers Table" (p. 123) doesn't have a "Target Running" penalty our team assumes the rules don't intend to penalize the defender in the situation you describe.

Other differences between the two tables of modifiers (Ranged p. 112, Melee p. 123):
Superior Position (Melee, but not Ranged)
Opponent Prone (Melee, but not Ranged)
Friends (Melee, but not Ranged)
Attacker in Melee (Ranged, but not Melee)
Aimed Shot (Ranged, but not Melee)
Attacker moving over Difficult Ground (Ranged, but not Melee)

Cannon Companion, page 86