Have you measured the battery voltage directly as a test?

Before rushing out for an alternator check all earths are good, particularly the engine earth strap, otherwise from what you describe it does sound as your alt may be struggling under load.

The regulator is integral to the alternator it is fixed to the back with two screws. The thin blue wire excites the field windings. There is also a suppressor mounted in the back with a small black wire.

Martin