So short and sweet, if I use coffin nail to discard zombies from my hand and have No swag in discard, am I still able to perform zombie discard?
The card text is mostly clear on this:
Draw a card
Discard any number of zombies in your hand. LOSE SWAG cards from your discard pile equal to the number of zombies discarded. (you may not discard more zombies then you have cards in your DISCARD PILE)
The bit in brackets on the card answers your question although it does leave a minor ambiguity: If you have only zombies in your discard pile then by the letter of the card you can potentially discard zombies without losing swag. (I think the bit in brackets should refer to swag cards just to make it simpler).