Looks like my problem was due to a shoddy soldering job. It looks like the drain and source on the mosfet were sporadically shorting out allow the current to flow to the relay. I cleaned up the spacing between the connections and it's been running trouble free now.
I thought my soldering iron was too hot, even at the lowest level, as I was having a hard time keeping the solder on the tip. Turns out the temp control potentiometer isn't doing anything and is sending full power to the iron. Guess I know what my next project is!
Awesome! Out of curiosity, do you even need the relay if the 12v is just triggering the 120v AC motor? It's possible you could just use the mosfet to trigger it plain and simple and remove 1 part of the equation.