Two thoughts. 1) 12v is not that much. I know its a 12v system, but at rest, it should be making more like 13v. While running, 14.4v. You could have a completely dead battery that still tests out at 12v. Its quite possible that your battery is dying since you've been taxing it with a failing starter.
2) The reman starter may test fine, but just because it spins and takes X amps to run with no load, doesn't mean that it actaully makes adequate torque. I had a similar problem with a barn fan. It wouldn't spin. I pulled the motor and it tested fine; 4.1 amps, 125v from the wall and after the motor, but when it was installed with the blades it drew 14 amps, didn't spin, and dropped the voltage to 80.
Try to explain that to Autozone... you'll usually get blank stares, but if you can get another one, it might be the problem. Also, ask to borrow a tool (for a deposit), but they have a little handheld computer for battery and charging system testing. It will tell you the actual cranking amps left in your battery. If its less than about 85% of original, it needs to be charged or replaced. It can't hurt to just go ahead and charge it, either but I'm sure you know that. If you know enough to test the posts, you know what you're doing
It could also be something as dumb as a corroded post on the ignition switch not giving enough juice to the solenoid for full engagement, a corroded post on the battery, or a bad wire to the solenoid.