Are you saying that the mositure you see in the basement is above ground level?
Assuming not, then you can likely remedy the problem with 2 easy steps. I did this at my house, and I have had no trace of moisture since:
1. Make sure that the ground (dirt) slopes away from the house. Ideally, this slope would equate to at least 2" per foot. If it does not currently slope away from the house, either add more dirt near the house, or simply move dirt that is currently several feet away from the house up against the foundation. You want to be sure that rain water is not able to pool next to the house.
2. Be sure that the gutters are not leaking, and add 3 or 4-foot diverters to the downspouts to carry the water away from the house.
If you still experience problems, then you may need to look at adding a French drain in the yard, or worst case scenario, adding a drain system in the basement itself. However, you want to do everything you can to keep the water out of the basement rather than trying to deal with it after it gets in.
As far as the drywall applied directly to the block, I would remove it, install furring strips or full studs, insulate, and then re-drywall.