Looking at your pics, my first thought is that there is ALOT of moisture there. (FWIW-We usually see that amount of moisture coming in from actual water infiltration as opposed to a moist air vent leak issue).
I would suggest that you:
1.) Purchase a moisture meter.
2.) Use it to test the areas of the sheetrock to determine the area of saturation/moisture penetration.
3.) Draw it out on the wall with pencil.
4.) Open up the wall by removing the sheetrock that you marked out.
5.) Remove all wet and damp insulation from the cavities.
6.) Test the framing members for dampness using the moisture meter. (Framing lumber moisture content should not exceed 19%)*
7.) Allow the "opened" areas to dry out. You could also use active means to dry, such as a blower or fan.
8.) Purchase a fungicide solution (not just bleach and water)** to apply over the affected framing members in order to kill any mold spores that may be present.
9.) Important: As you are opening up the area: Look for the traces of water stains and water. Water/moisture leaves a trail. This is the best and easiest way to determine where the moisture is coming in from. If it is coming in from the dryer vent alone, then it would be evident by the moisture stains being isolated to the general area of the vent hole going into the wall. If it is coming through another area (like the outside/grade issue) - then you will be able to trace a moisture line into the home and towards the saturation point that you found in the wall on your initial inspection.
9.) ONLY AFTER YOU DETERMINE the positive location and cause of the water intrusion, and air out the wall cavities properly, should you then install new materials and close the area up.
** 15. Do not use chlorine bleach to kill mold or disinfect moldy areas. Bleach is not an effective or lasting killer of toxic mold growth and mold spores on and inside porous, cellulose building materials such as wood timbers, drywall, plasterboard, particleboard, plywood, plywood substitutes, ceiling tiles, and carpeting/padding.
19. Re-spray twice the cleaned out area with another wet spraying of an EPA-registered mold fungicide or an effective mold home remedy to kill any remaining, living toxic mold spores or mold growths. - http://www.buildings.com/Articles/detail.asp?ArticleID=2285