It might be your input connections list. Give this a shot:
Select your master shape (The one the blend shapes are applied to) and right click to bring up the options. Under Inputs, go to "Complete List..."
Now, I am assuming your character has joints and is bound, which causes your symptoms.
The list that appears should look like :
Blend Shape (blendShape1)
Skin Cluster (skinCluster1)
Tweak (tweak1)
Ignore the tweak. Our beef is with the first two. What they are are deformation controllers. They tell Maya which deformations to do first, and which to do last (going from bottom to top). So each action is evaluated seperately, and in order. What is happening is that your joints are deforming your object, but then your blendshape node (the one on top) is overriding these deformations to achieve its goal. Simply grab the Blend Shape item in the list, and bring it below Skin Cluster item. What this will do is tell Maya to first calculate the appearance of the Blend shape, and THEN apply the joint deformations.
You should notice an immediate change in the reactions of your model. Hope this helps!