Whirlpool model LSR7010PQ1 drains water but will not spin out the rest of the water left in the clothes once cycle is done! What could the problem be? It is costing a small fortune at the laundromat!

There are several things it could be and you will need a digital multimeter to properly diagnose this problem. You have a spin solenoid and an agitate solenoid and either could be open or have a stuck plunger. With the meter on "ohms" you should read between 200-700 ohms if the solenoid is good. You also have a clutch that could be bad. Here are some manuals to help you diagnose it: https://dl.dropboxusercontent.com/u/8010163/Understanding%20automatic%20washer.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool-Kenmore%20washers.zip
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Washers.zip
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Direct%20Drive%20Washer%20Repair%20Ser vice%20Manual.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Washer%20L-61%205th%20Generation%20Solid%20State%20Electronic %20Control%20System%20Service%20Manual.pdf
Make sure to get a good digital multimeter (DMM) at Home Depot, Lowe's, or Radio Shack.

This model doesn't have a spin or agitate solenoid; Most likely cause is a broken motor coupling located behind the motor connecting to the gearcase; clutch could be slipping if coupling is okay..