how significant is the effect of high loads on servo accuracy?
First, let's get straight that you are actually talking about small "hobby servo motors".  The word "servo" has a different and general meaning having to do with a control loop method.

I didn't look at your particular hobby servo motors, but if they are like most others the unit includes a motor, position sensor, and some kind of electronics that performs servo control such that the motor is driven to a particular position.  The usual means this is communicated to the electronics is by sending a pulse between 1 and 2 ms in length, with 1 ms indicating one end of travel and 2 ms the other.  These pulses need to be repeated usually every 20-50 ms for smooth operation.

Due to the fact that these units perform closed loop control of the motor position, the power to the motor will be dynamically adjusted as needed to get to and hold the indicated position.  The motor will automatically be driven harder when it is off position, or something is trying to force it off position.  Depending on the exact servo algorithm, small deviations may not produce as much torque from the motor initially, but it may produce more and more force over time if a small position deviation persists.

The system is not about moving X degrees, but holding the motor at X degrees.

If these are typical hobby servos, you don't implement a PID loop to control them.  That's what the servo controller in the unit already does.  You simply send it commands what to do and it will drive the motor as necessary to follow those commands within its limits of speed and torque.