Your city location poses a problem. The rain has ruined the drywall. It may only have been mildewed and could have been washed and installed- but not now. Fungus spores are EVERYWHERE-- fact of life. If I was educated, I would say they are omnipresent. They do not grow at 50% humidity or less. So even new sheets will soon mildew in damp environments.
The EPA has insane rules about the disposal. Gypsum is an excellent soil conditioner and I just break it up and scatter it in my fields around my house in this rural area. The EPA says that paper must be removed first. It biodegrades here and also conditions the soil.
Edit:
"I carried it outside of the house carefully, and now it is leaning against an outside fence outside our house waiting for me to figure out what to do with it. It got rained on this weekend so hopefully dissipated some of the mold."

We understood this the first time.

This post was edited by bus_driver on Thu, Feb 28, 13 at 19:46

It got rained on this weekend so hopefully dissipated some of the mold.

No. Water is what mould needs to thrive!

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Businesses in NYC must recycle drywall. It looks like residents can put it aside in bags as construction waste. Call 311 to be sure.

Interestingly, some municipalities, such as Toronto, Canada, provide a few convenient locations for small dropoffs such as yours, where the materials are then recycled.

As for the dangers, not likely unless someone has special environmental sensitivities.

Here is a link that might be useful: Centers for Disease Control on Mold

This post was edited by worthy on Fri, Mar 8, 13 at 10:00

the drywall is now OUTSIDE, not in our basement, so getting rained on is kind of irrelevant to our interior at this point, and I've confirmed that NYC doesn't have special requirements for disposing of drywall, even with some mold growing on it (go figure), so off it will go

"NYC doesn't have special requirements for disposing of drywall, even with some mold growing on it (go figure)"

NYC trash is barged to other locations and buried in landfills for the most part.

you have shared your problem in your post .Don't avoid it for long time .I am not sure but may be this problem of heavy rain in your region.could you tell me about rainy season in your city?

If the moisture level in your basement is high enough to cause what you believe is a serious mold problem with drywall, what about the effect on all the other materials in that area?

Thanks to discuss with us.you can use dehumidifiers to detect moisture in hidden parts.I think it really will help you.