The yaw damper was the rudder axis of the autopilot on the KC-135A. I can't remember any discussion of this helping to coordinate turns. The standard practice was to keep the rudder autopilot engaged when flying manually to prevent the nose from swaying side to side. There may be some vaildity to your coordination theory.
The KC-135R has an Engine Failure Assist System (EFAS) for take off, which will automatically deflect the rudder in the event of engine faliure on takeoff. I believe, but can't exactly remember, that the rudder axis of the auto pilot was upgraded at the same time. And a true yaw damper added?
Man, I'm getting old. I used to know this stuff, chapter and verse.
A yaw damper was probably the first electronic augmentation to flight controls ever designed. It's a fairly simply analog rate detector that will input a few degrees of rudder opposite sensed yaw. While it's designed specifically to counter dutch roll (the "swaying side to side" you mention) it also provides some (limited) automatic correction for coordinated flight. While jets don't have to worry about all those nasty effects from that spinny thing up front the use of ailerons will still have a tendancy to produce adverse yaw (yaw away from the turn) because the upgoing wing is producing additional lift and therefore increased induced drag while the downgoing wing produces less lift and less induced drag. It's this differential that will yaw the plane away from the roll and turn. A yaw damper will counteract the yaw within its limited authorities and provide a coordinated turn, again this is within limits.
Yaw dampers evolved into Stability Augmentation Systems (SAS) with damping in pitch, roll, and yaw but their authority is still fairly limited in the amount of control deflection they can command. Over time these analog systems have been replaced by digital computers and became CAS and TCAS type systems with lots more control authority and new features like aileron/rudder interconnects and then, eventually, pure fly-by-wire. The newer systems are smart enough that they can provide the right control inputs regardless of what the pilot does.
For instance, the F14 had only a 1960's era analog SAS and when pulling greater than 17 units AoA the pilot had to know not to use lateral stick to roll but instead use the rudder. At high AoA a pure aileron roll produced so much sideslip that the airplane would actually roll opposite of the stick input...it would start in the correct direction but the induced sideslip would blank and stall the upgoing wing while the sweep of the downgoing wing dramatically increased its lift and the airplane would snap roll the other direction, violently if you were fast enough. This was the negative side to dihedral effect and swept wings and the SAS didn't have the ability to counteract the problem while it did help some. However, using rudder instead of stick yawed the airplane into the desired roll direction producing sideslip from the opposite side and dihedral effect would then roll the plane nicely. (This is the same cause/effect as the rudder roll you saw in the T38). The F18's fly-by-wire system is smarter. It has two flight control computers that receive the pilot's input from the stick/rudders. The computers look at the flight conditions and decide what commands to send to the flight controls. At high AoA it will washout aileron and replace it with rudder even if the pilot just jams the stick to the side. The result was the same as in the F14...the F18 pilot just doesn't have to be as good since he only has one vote in three.
