Tearing out the drywall yourself and taking it to the dump yourself it always going to the be cheapest. When we have construction debris it can usually be broken down to fit in black contractor bags, that either can be collected as part of our weekly trash pick (actually cheaper than going to the dump on our own) as we almost always make less than we are allowed; though sometimes we have to spread it out over a couple weeks. Or the dump is not out of the way when we go to the grocery store and the like, so taking a bag or two at a time in a non-truck is also an option.
Honestly drywall can be broken and taken out in any size pieces so your husband could do it on his own. In terms of health, pretty much no mold should make it past a properly fitting N/P 95 or 100 mask, but invest in a decent 1/2 or 3/4 face mask (and with god goggles, in not just a full face) and not one of those paper masks (they almost never fit properly).
Be prepared for the mushroom factor (perhaps literally). If the roof is leaking and the drywall is moldy, then there is a good chance that everything behind the drywall is too; that area will be moist, from the leak, warm, from what I assume is a dark roof, and have organic material, the wood framing, for the mold to feed on. Depending on if it is insulated and if so what kind of insulation it could be messy/moldy. You may find that whatever is behind the drywalls needs a good treatment to kill off the mold.