Steering all of a sudden starts, now forgive me, "goes back and forth, by itself..." when i came to a stop, full stop with foot on brake, the steering wheel "goes back and forth" by itself...seems to happen randomly...

If you are stopped and can turn the wheel back and forth, it is not the power steering.

Sounds like the transmission

Check this link for location.

Along side the battery , infront of the brake fluid holding tank
http://www.carcarekiosk.com/video/2010_Lincoln_MKZ_3.5L_V6/transmission_fluid/check_fluid_level

Your tires need balanced.

When the ABS comes into play from heavy braking you will feel a pulsating of the brake pedal under your foot. This is the brake pressure, on off on off as the wheels stop start stop etc. The ABS will not control the vehicle direction but by allowing the wheels to turn under brakes and not lock up you have control of the steering and the direction the vehicle goes.The ABS will not stop you quicker and in fact on ice or loose surface the distance to stop will be increased. As I said all the ABS does is give you steering ability under brakes.. Take the vehicle to an accredited brake shop and have the ABS tested if you feel that it is no operating properly

Have you checked the power steering fluid level? The reservoir for the fluid will be at the front of the engine, where the belt runs the power steering pump. If it's low or out add a little power steering fluid till full, then start car and slowly turn wheels back and forth from stop to stop-not too ******* the stops (when the steering won't turn further that way...that's what I mean by stops)....Turn wheel one way full travel, then reverse and turn the other way. Check the fluid level again, and add as needed.
If fluid is not low, the belt may be too loose to make the power steering pump work, or you may have a bigger problem that needs professional attewntion.

Yes it is very possible.Lift the hood ,have someone stand to one side and watch the engine.Put left foot on the brake,and press the accelerater with the car in DRIVEand bring up the revs slowly.Then repeat with the car in REVERSE.The engine should not move more than an inch or so in either direction.I have seen bad engine mounts let an engine move back and forth six inches by doing the above .BE CAREFULL!!! make sure your foot is down on the brake 100%. Good Luck

try disconnecting the battery briefly and then reconnecting it. I had this happen to me on my 04 9-3

This condition indicates warped rotors. You will need to remove the rotors and have them turned on a brake laythe. Since you will have the calipers off, go ahead and replace the pads...its an inexpensive maintennance item. Good Luck, Hope this is helpful.

It will probably just sit there when you start the engine, then take off like a rocket when you touch the steering wheel? The wheel will take off in the opposite direction of the way you tried to turn the steering wheel. (you start to turn the wheel to the left and it rips out of your hand and goes right until it slams the full right stop, then rockets back to the left until it hits the full left stop and shoots back to the right again and repeats this process until the engine is shut off.)
If this is the case, you have the hoses from the control valve to the power cylinder reversed.

make sure its the right key,if so try a spare key,,key may be worn NAPA has wet grafite spray try that none of thi works ,try shaking steering wheel bac and forth while trying to start it,make sure its in park good and foot is on brake

check your motor mounts by holding your foot on the brakes, set the E-brake and have someone watch the motor as you shift into reverse and touch the gas a little bit do the same in drive if the motor moves allot back and forth you need new motor and possibly transmission mounts good luck and please rate -jeff