Definitely moisture issues, possibly due to lack of insulation, hot humid summer conditions or maybe both. Conditioning of an uninsulated space will likely just make it worse. Trying to cool a space with no insulation causes the warm moist attic air to condense on the top side. Heating the space will cause the warmer moist air inside to hit the colder ceiling and condense on the bottom side, either way the drywall wicks up the moisture.
Without conditioning the space at all, the warm humid days can also take a toll on drywall. Drywall simply can't handle a lot of moisture, the screws or nails will begin to pull through over time as the drywall softens. Drywall is not a great choice for unconditioned spaces but is required for fire code, especially in an attached garage. Yours probably just didn't get enough screws or nails to survive your hot, humid summer conditions. Depending on the condition of the drywall you might be able to add screws and pull it back up. You may want to add a plaster washer on each screw and you still need to work along the joist tightening the whole row of screws to pull the sheet up a little at a time or the screws may just pull right through.
Laying 1x4 (or 2x4) on the drywall, perpendicular to the ceiling joists, on 2' center will pull it back up and offer support. You'll need to screw through the boards and drywall into each ceiling joist. The boards will then support the drywall and you can blow insulation on top. You could then choose to install liner steel over top and fasten that material to your new wood furring strips. If that is what you prefer, then I'd definitely go with 2x4s. After the drywall is supported you should blow a layer of insulation on top.