Does an increase in load current change field winding current?
Depends on the configuration of the field coils. A series field coil passes the same current as the load. This current increases field strength and causes the generated voltage to increase under load, further increasing current through the field coil. 

A shunt field coil is wired in parallel with the output, so its current is proportional to output voltage and decreases as the load increases (since load current causes a voltage loss in the armature resistance). 

A compound-wound generator has both series and shunt field coils. The series coil is sized to maintain output voltage under load, so the shunt coil draws (almost) constant current.