Visa does not cover my stay duration

Hello,
I was issued a Chinese entry permit using a 3rd party agency (J2 category, since I'll be there for work initially), and it was only granted for 12 days. It's already been put in my passport and finalized. However I already have prepaid lodging and other activities for an additional week after the duration expires. What are my options to extend my stay? Can I go to the Consulate in NYC tomorrow and get this extended or stack an additional one? I leave for Beijing 6 days from today. I am hoping I don't have to cancel the 2nd part of my trip or leave the country and apply for a new one and come back.

Is it overstaying, and willing to pay the fees an option?
Please help! Thanks!