Patrick,
I'm sorry to hear you're experiencing the same thing we did. I wish you luck. Here is an update on my issue:
I had pretty good success in getting Ford to respond to me. I was in email and phone contact with the regional rep who seemed eager to help and used all the right customer service phrases. Unfortunately, it came down to money. The dealer would not explore the problem further than what could be examined superficially. On only ONE occasion, did the car produce the characteristic smell during a test ride with the service manager. I was so excited because he had the same reaction I had. He too couldn't put his finger on it...it had a sweet component, but was also a little thick/oily smelling and stuck in your throat and after 10 seconds, was kind of sickening. I really thought this was the day it would be fixed. But no.
They said that in order to really diagnose the problem, the whole dash would have to come out and since it was 8+ hours of labor...just to explore what MIGHT be wrong...they were not willing to do it. It took a long time to get the service manager to admit that and I had to ask some specific and leading questions.
I am completely dissatisfied with Ford's service policy - especially when the car is less than a year old and had less than 7K miles. By the way, DO NOT let the service adviser talk you into getting an ozone treatment. Not only did the treatment make the car smell worse (different, but now all the time), it made the interior dirty, most noticeably, the headliner with circular brown/black splotches. This was done against my will, because as you're probably aware, the smell IS NOT coming from the interior or plastic outgassing. They just did it anyway.
I too have concerns about whether the smell could be harmful and since this car was for my wife, I didn't want to take any chances. I ultimately traded the car for an Explorer in January.
My issue wasn't so much with the Edge itself, but with Ford, so I'm embarrassed to say I stayed with Ford for the replacement. But the Explorer was the only real candidate in our price range. I felt I was taking a gamble in hoping nothing quirky would go wrong and having to enter another frustrating ordeal at the dealership. So far, so good.
Trading a relatively new car is usually a costly experience, but I can offer no better advice to you than to get rid of the Edge before the frustration gets the better of you or the smell makes you or your family ill. Good luck.
Sincerely, Art