Why does it feel super awkward and uncomfortable when the blood rushes back to an extremity that has "fallen asleep"?

Body parts have sensory receptors which send signals to the brain, notifying the brain of what's going on in that part. For example, there are touch receptors on the surface of all skin which tell your brain when you are touching something. There are also pain receptors whose sole purpose is to tell the brain when the body part is being harmed, telling the brain to feel pain.

A body part "falls asleep" when it doesn't receive adequate blood flow. Oxygen and nutrients from the blood are required to power all the sensory receptors. When a body part is "asleep" it cannot power the touch receptors, hence the feeling of numbness. When blood flow is restored, all the sensory receptors are rebooted and start working again. The pain receptors start firing spontaneously because they are confused. That is why you feel pain or discomfort when recovering from this experience.