Will this moisture problem persist after drywall is installed?
You need vapour barrier (the plastic you put up before drywall) and air barrier.

To count as an air barrier you stop air flow by using sheathing tape on:


OSB joints and edges
Vapour Barrier (this is most common)
Tyvek Housewrap seams (becoming more common)


Some people think that you only need a vapour barrier or that the vapour barrier is also an air barrier. This is not true. You can make it an air barrier by sealing the joints and seams with tape and mastic, but otherwise it is only to stop Vapour Diffusion and not Air Flow.

Long story short: you had condensation because you did not have a proper vapour barrier, air barrier, or both. Since you seem to have a VB built in to your insulation, all you need to do is tape the seams and seal the edges at the top and bottom plate to make an air barrier.