Question: if a character were to use a melee weapon in their off-hand while wielding a pistol in their on-hand, would you require them to use the off-hand <insert melee weapon here> rules or simply the normal <insert melee weapon here> rules. The major difference being that one provides you the complete skill rating in dice for rolls while the other only provides 1/2. Thoughts?

Full off-hand skill with a TN modifier between 1 and 4 as per rules of C&C ...

My take on the TN mod value:
Non-ambidextrous characters usually getting the full +4 mod, since the edge provides -1 per level ...

Alternatively with ambidexterity rules:

Pistol as improvised weapon with club skill (default where appropriate *with appropriate TN modifier) plus halved off-hand skill (with no requirement of actual off-hand skill at ambidex levels 3 and 4) ...

I don't penalize them, so long as they use only 1 at a time.

If the weapon is in the off-hand and the character isn't ambidextrous, then of course he or she should use the off-hand weapon skill.

So would they need "Off Hands/Fists" to punch someone?

No, because the 'unarmed combat' skill involves using both fists to beat someone down. If it didn't, you'd get that +50% bonus just for not only jabbing with your right. Duking it out with someone and trying to weild a weapon in your off hand are different.