Yes you can have foam insulation sprayed directly to the backside of the drywall.
Before doing that though, I would look for other causes. Are the bathroom vents or kitchen hood venting into the attic? This is a BIG no-no. This causes all the moisture to go straight into the attic. If this is the case, odds are if they are properly vented (directly to the outside) the moisture problem will go away.
One other thing to look at: Are the soffet vents plugged up with insulation? Sometimes when they blow in insulation, they are not careful about making sure they are left open.