I filed an amended form and now owe $0.00. Do I still have to send the first amount and wait the 16 weeks for it to be refunded or do I not send anything?

Just to be sure what you are asking, your original return had a balance due, say it was $100.  You have amended your return and it now shows a refund of that $100 that you owed on the original return. 

So, your question is do you need to pay the $100 and then file the amended return to get it back.

As long as you file the amended return before the due date of the original return (and thus the due date of the $100 balance due), you do not need to pay the balance due.  You have corrected or changed the tax return that showed the balance due in the first place.