2002 Ford Fiesta electrical and charging problems

Hi there, I have the following issues with my mk6 fiesta and haven't s clue where to start, the car headlights brighten and dim whilst driving, causing other drivers to flash me, almost like they are flickering, the heater blower motor also blows slow then fast in line with the lights, as though the power is being drained, the higher the revs the faster the cycle of problems, then the battery light also occasionally comes on, but the problems seem to stop when this happens and re-occur when the battery light goes back off??? I've changed the alternator as the old one appeared to be giving overcharging readings, but this hasn't rectified the issue,😢 the alternator plug also appears to be clean and intact. Any advise or ideas will be greatly appreciated 😀