Why are my headlights getting brighter and dimmer and my AC going faster and slower in my Ford Focus?

This issue had started about 2 months ago my battery died and so I replaced it I had my alternator tested and it tested good but I still have the issue with my headlights getting brighter and dimmer and my AC fan speeding up and slowing down even while idling

When the a.c. Kicks in it uses enough power to dim the headlights. If you rev the engine a little with the a.c. On the lights should get a little brighter.

Check the voltage regulator. Modern alternators have a voltage regulator built into the alternator. You might have bad wiring (burnt?) to the alternator (where it connects to the voltage regulator). So, with the battery out of the car, you can use an ohm meter to test the wires to the alternator.

You're AC is going faster and slower. That's probably the blower motor fan you're talking about.
A bad ground connection can cause that. A bad alternator can also cause that.
I'd start with cleaning the battery posts and cable ends.
Run another ground wire from the motor to the alternator housing.