Dead band and Gear lash
I am by no means an expert in servos, but I have some experience, so here is my shot:
There will always be a little gear lash, a servo with nylon gears should have less than one with metal gears.
Also note that servos have a deadband, think hysterisis, where the pulse must change beyond the dead band. A deadband of 1usec means that the pulse width must change by more than 1usec before it will change at all. So, you have to nail the spot first time.
I think you may need to go with a stepper. I’m impressed that you got within .1 degrees! Also, I don’t understand this continuous rotation servo you have, normally one that has been modified to rotate continuously no longer knows where it is, just how fast it needs to travel.