I just noticed that the 4.0 code differences are too much anyway.
A simple workaround for 3.8 is setting AnimationState.Data.DefaultMix
to the desired time:
var savedDefaultMix = skeletonAnimation.AnimationState.Data.DefaultMix;
skeletonAnimation.AnimationState.Data.DefaultMix = 0.0f;
skeletonAnimation.AnimationState.AddEmptyAnimation(0, 0, 0);
skeletonAnimation.AnimationState.Data.DefaultMix = savedDefaultMix;
This leads to the proper delay being used.
However, after reading your initial posting again, I assume that you are facing a different issue.
Actually after mixing out all anmations there should not be any remaining pose applied. Anyway, you can clear the bone locations by calling
skeletonAnimation.Skeleton.SetBonesToSetupPose();