Spaghettification while still in orbit THIS POST does a pretty good job of giving an idea as to what the differences in forces becomes as you approach a black hole.
For a small enough black hole, you would reach the dangerous tidal force distance before you reached the event horizon, which means you could have a stable orbit around the black hole near that dangerous distance. So to prevent from getting torn apart, you could lay down like a plank, facing the black hole while orbiting in the danger region, the differences in force on different parts of your body would be uncomfortable but survivable. But if you suddenly extended an arm towards the black hole, that arm would have a dramatically different force acting on it.
Assuming you are far enough so it doesn't quite rip your arm off, would this sudden change in forces acting on your body change your orbit? My initial thought was that since you haven't changed your momentum or center of mass at all, your orbit would remain the same. Then I realized that since the gravitational force is proportional to 1/r^2, that means that even though your COM is the same, the total force acting on your body has increased by stretching your mass out perpendicular to the gravitational body, which means your orbit would change right? How would it change your Apoapsis and Periapsis, if you started in a perfect circular orbit?
As a follow up question IF you can change your orbit this way: Could you build a machine(and thus higher tolerances than any human) that could orbit a black hole, and just by moving its mass around at the right times in orbit, change its orbit dramatically?
 A: If you did extend your arm out, you would have indeed changed your angular momentum, and for angular momentum to be conserved your orbit would change slightly in the opposite direction. In addition (I am going by your theory of "partial Spaghettification,"), if your orbit was not thrown off by the gravitational pull of the black hole(which it would be, I will address this in a second), you would have still changed your mass on one side, so for momentum to be conserved you would move slightly in the opposite direction (if you want to go left in space, you would throw cargo off to the right, but practically speaking, there are better methods for navigation). However, if your arm is attached to the rest of your body, you would get pulled in along with it (assuming that the highly fictional, theoretical situation which you depicted is actually going happening), like if somebody was holding your hand and pulling you toward them at a certain force. Your cells are still attached to one another, and they would therefore exert a "pulling" force on one another as the result of the force the black hole was pulling you in with.
