Thanks to those who posted replies. Let me preface my dilemma by mentioning that my house was built in 1961 on a slab. Ouch, never again. 1st time homeowner and didn't know any better.
At any rate. I still believe I have some kind of mechanical blockage. I went and got the snake machine today and ran it as far as I could go until the bends beneath the tub stopped me, and I retrieved zero hair.
Again, our drain worked fine and quickly two days ago, then bang, completely stand still water. The face plate and stopper came out in one nice piece. This just doesn't make since to me. I think I may have to bite the bullet and call the plumber.