The flex plate should bend slightly, hence the name "flexplate". The convertor nuts won't have an effect with the movement of the flexplate since it is rigidly mounted to the engine's crank. If the convertor nuts were loose, the convertor would move back and forth, but the flex plate would stay still. You'd have some interesting knocking sounds at idle if the nuts were loose.
Move the flex plate in and out again, and see if your harmonic balancer is moving in and out too---if so, then your engine's thrust bearing is shot. I have seen cars at the shop with a lot more then an 1/8th of an inch of thrust clearance, and they drove normal.
My Ford book recomends .004-.010 thrust clearance with a max of .012. If your measurement of .062 is accurate, then it sounds like time for some bearings, or complete engine overhaul.