You are correct in that during a typical spin, the first that happens is that the pump drains the standing water from the machine. That is the first noise you will hear and should be able to see the water level drop comletely within 1-2 minutes. Then while the pump CONTINUES to run, the basket will move back and forth as the motor controller attempts to determine the load conditions by monitoring current draw before attempting to enter a spin. Then the machine will enter a low spin, still monitoring the draw, then the high spin. So it sounds like the machine is operating correctly. Now, as far as the slime, this is MOST LIKELY caused by a buildup of soap, fabric softener and other chemicals used in the wash. If you are not already, you MUST use a good quality "HE" (high efficiency) soap in this and all front loading machines, and ONLY about 2-4TBS MAXIMUM of that soap for a FULL load. That amount WILL wash your load completely.
Now , as far as what may be causing the sopping wet clothes, if the machine is spinning and FOR THE MOST part, draining, you probably have some residue and crud built up in the tub-to-pump hose , the coin trap, and the impeller housing of the pump. you MAY also have a broken blad on the impeller or loose impeller shaft. Look at the diagram of your machine below :
Item 40 is your drainage filter and is located inside of item 41 which is the tub-to-pump hose. Removing item 48, the spring clamp for the hose will allow you to inspect the coin trap, the impeller housing for the pump and give you access to the filter which you can remove and clean or at least inspect. This filter is not the easiest to remove without removing the entire tub-to-pump hose, but can be done. If you had a bunch of slime at the drain hose, I know you will even more slime and probably debris in this area as well.
The BAD news is that Frigidaire, in their infinite wisdom have made access to this MUCH more difficult than ALL of their previous models because they eliminated the front access toe panel in the front, so you must either remove the ENTIRE front panel to check OR go in through the REAR access panel which gives you much less room to work on the FRONT lower areas. Dont ask me why they did this, but they did.
Now as far as shutting off with 6 minutes remaining on the timer, this could add a whole new dimension to the diagnosis, especially without any error code. You said you have watched the machine enter a high spin before, but if you are saying now that it WONT even enter the high spin, this COULD be caused by a either a failed motor controller OR simply that the machine is not satisfied that a safe spin can be entered into, in which the machine WILL stop after the "checking" phase of the spin (back and forth with pump running) and will not generate an error. Check this with NO load and see what happens.