How does the speed of a synchronous generator changes with load?
Here is a description of the overall picture based on the assumptions mentioned below.

Assume the generator is a 3-phase generator driven by a prime mover. It supplies a load but is not connected to a grid with other power sources. The connected load determines the armature current and power factor. Let us assume that the generator’s automatic voltage regulator (AVR) adjusts the field current to keep the generator’s terminal voltage at the rated value. Let us assume that the prime motor is an internal combustion engine.  Let us assume that the engine’s speed regulator adjusts the throttle to keep the engine at the rated generator speed.

The electrical power delivered by the generator is given by Pe (watts) = V (volts) X I (amps) X pf (power factor) X 1.732 (square root of 3). The mechanical power provided by the engine Pm = Pe + losses. Pm = Torque X Speed X a units of measurement constant. The torque required to drive the generator at the rated speed is determined by Pe + losses.

If the load changes, current will decrease and the power factor may either increase or decrease. The terminal voltage will increase because of the change in voltage drop across the generator’s internal impedance. The AVR will change the field current to maintain the rated terminal voltage. The torque required to maintain the generator speed will decrease because the power will have decreased with very little voltage change. The speed will tend to increase, but the speed regulator will change the engine throttle setting to maintain the speed.

At the new operating point, the speed and voltage will not have changed. The current, power, torque, torque angle and fuel flow rate will have decreased. The power factor and field current could either increase or decrease.

Summary of the essential points

A change in load is by definition a change in armature current. Since nothing happens instantaneously that would change voltage very much, a change in load must be a change in electrical power drawn. By conservation of energy, a change in power drawn must cause a change in torque produced by the generator as a load on the prime mover. That could be illustrated by examining the current,  magnetic flux and force interactions. A change in torque presented to the prime mover will result in a speed change. Regulating systems react to the change. The magnitude and duration of the change can be determined by detailed dynamic mathematical analysis analysis of the entire system.