No, you're not stirring enough, and no, there should be no water left.
You should be able to do your mash at 1.25-1.5 qts. per lb of grain, stirring many times during the mash.
Then drain your first running and measure the volume.
Whatever the amount that is still needed for the batch size, that's how much more sparge you will add for the second running (or divided for multiple step sparge).
There is valuable sugars left in that liquid. That explains (one way) you have low efficiency.
Keg #1: Brown Bomber Bourbon/Oak Vanilla Porter
Keg #2: The Bollocks ESB
Keg #3 Lawnmower APA
Bottled: Mjolnir Mead, Lazy Daze Hefe.