I am trying to weight paint my model but no matter what weight I paint or assign to the verts, I can't get the skin stretching effect I want. They all act as if they have full weight and only the verts next to the ones I painted stretch.
A weight of 1 (red) means, that your mesh will react 100% to the bone movement. A weight of 0 (blue) means, that your mesh will not react at all.
So, if you want a more natural behavior, select the lower vertices of the head and assign a weight of 0.5 (green). These vertices will only be partially affected. You can go further and have several transition loops: Assign it like this to the top bone.
And assign reverse weights to the bottom bone.