I agree 100% with the advice to let it air out as well as to keep it out of direct sunlight.
This thread reminded me of something i thought of a while ago. If you have some sort of contamination, which is a problem when you do X, then doing X' should eventually get rid of the problem. In your case sniffing air that is coming off the inside of a cooler is a problem due to cooler "contaminating" that air (X). So by getting allot of air through the cooler (X') you in effect wear out the problem.
Here is an extreme case scenario: if i am trapped on an empty island and all i have to drink out of is a glass that had oil and other nasty stuff in it, i am going to do my best to wash it out with clean water (assuming i do not have soap or other surfactants available) which will be a huge hassle to get it completely clean, but if whatever is left in that glass i cant get out with water, then the water i will be drinking from it wont get that nasty stuff in my body either. I am NOT suggesting that is is safe to drink from your chemistry set glassware or that you should drink from oily glass, but my point is that if doing X (drinking water from glass) is a problem due to contamination, by doing X' (washing glass out with water) will eventually get rid of contamination. In your case airing the cooler out is X'.
I'd also suggest placing a fan in front of the cooler with back of the fan directed at the cooler, so that fan is sucking air out of the cooler. By reducing pressure in the cooler you may accelerate the scent inducing molecules to leave the surface. Shoot, you can probably just stick a vacuum hose in the cooler and tape the lid, so vacuum is sucking air out of the cooler.
EDIT: don't let the difference in advice of blowing air in vs sucking the air out confuse you. It will all work the same, it comes down to air replacement cycles as stated above. The reason why i suggested vacuum hose sucking air out is because this should reduce pressure inside the cooler, hopefully accelerating the rate at which scent inducing molecules leave plastic. How pronounced this effect be as compared to blowing air in, i can't tell you.