There are two enemies to the carved pumpkin, mold (the fuzzy stuff) and moisture loss (the wrinkly, shriveled look). To fight moisture loss, smear petroleum jelly on all the cut surfaces right after carving. Some folks coat all the internal surfaces but I just focus on the ones that are part of my design. The thinner and more intricate the pattern, the faster this drying out process will happen.
Even with the cut surfaces coated, your carved pumpkin will eventually start to shrivel, looking a bit like my dad after he takes out his dentures (photo not included!). Soaking your pumpkin in a bucket or sink of water for a few hours or overnight will plump it back up. I have used this trick repeatedly with the same pumpkin to extend its life.
Here is the really cool part: to fight mold, simply add a few tablespoons of bleach to the water while your pumpkin is soaking. Mold also likes heat to grow. We live in North Dakota, so finding a cool place to display our pumpkins is never a problem. For the rest of you, consider popping your pumpkin in the fridge over night or whenever you aren’t displaying it. Just like other fruits, being in the fridge makes your pumpkin last longer.