Be careful with using automatic test mode to diagnose a spin problem as it won't let the washer spin up to full speed. Manual test mode would be more useful for this. Also, you can try clearing all the errors in memory, then passing all the steps in manual test mode one after the other and see if an error comes after all tests.
When the unit is shifted for spin mode, when you turn the tub by hand, it usually turns the motor. The agitator also seems to move directly with the tub as if they're stuck together during spin, they can move separately only during agitation.
Maybe also the new splutch wasn't installed properly, usually you can spin the tub by hand just by turning the pulley under the gearcase.
Usually during agitation, it's only the inner part of the shaft that is moved by the motor. Then during spin, both parts of the shaft move together.
You might better see what's happening with the agitator plate and spin tub removed. The plate is only attached with a 7/16th inch bolt, and it's only a clip holding the spin tub in there.
You'll only have to remove the tub ring so you can slide the spin tub upwards. Then you can see the splines for both the agitator and spin tub going out of the gearcase. If the splines are damaged or the tub is not attached firmly, it can explain part of what you're experiencing.
With the spin tub removed, you might see this more clearly, just tilt the top of the washer back on it's hinges even with the lid locked.
Just for your reference, here's the exploded view and part list for your model:http://www.appliancepartspros.com/parts-for-whirlpool-wtw5000dw1.html
If the tub spins when you are turning the pulley by hand, but not when it's the motor, maybe there is a problem with the basket speed detection, causing the board to not engage full speed spin. The sensor is optical and part of the shifter, so if an obstruction causes feedback to the board that is too different than what is expected, it might prevent full speed spin or abort the cycle.
The first symptoms of a gearcase failure usually is that it gets noisy for these models as the gearcase mostly contains the bearings and all shifting mechanism is external.
Also, the gearcase is an exepensive part so you don't want to replace it unless necessary. Hopefully this will help you figure out where the problem could be.