Sounds like you're along the right lines. I have reconditioned 5 of these but have never seen this problem. Most people have the opposite. I would love to have a look at that one. It doesn't seem like a problem from a stretching cable, more like a sticking cable. If it had stretched it would stop as the pulleys in the side of the unit "should" all be spring loaded to disengage. So I would hazard a guess that you need to get the side off and oil the whole unit.
The inability to push when it's off is common but again is solved with oiling the bearings and greasing the cogs. Get your Allen key out and take out the 4 bolts on the large left side of the mower. Oil/grease everything except the two drive belts. The cables engage/disengage pulleys to the drive belts and are spring loaded. Hope this helps.
MowerMan
August 2011