2013 MAZDA MAZDA3 Electronic Stability Control Esc Problems

I HAVE A 2013 MAZDA 3I 5-DOOR HATCHBACK WITH APPROXIMATELY 61,000 MILES, 90% HIGHWAY DRIVING TO AND FROM WORK. AFTER HAVING THE BATTERY REPLACED (ORIGINAL) EARLIER THIS YEAR, I THEN NOTICED THAT MY HEADLIGHTS WOULD FLASH INTERMITTENTLY, AT THAT TIME. AS THE YEAR PROGRESSED, THE HEADLIGHT FLASHING, WHICH PRESENTS ITSELF ONLY UPON BRAKING OR DECELERATION, BECAME MARKEDLY INCREASED AND BRIGHTER, INCLUDING THE DASHBOARD INFORMATION SYSTEMS WINDOW INSIDE. I WENT TO AN AUTHORIZED MECHANIC, WHO FIRST PUT MY CAR ON DIAGNOSTIC, WHICH SHOWED THAT THE ALTERNATOR WAS VERY WEAK/WORN. SINCE I HAD EXTENDED WARRANTY, THROUGH A PRIVATE COMPANY, BUT ONLY UNTIL 60,000/5 YEARS, I RUSHED TO MAKE TIME FOR THE MECHANIC TO RECEIVE AUTHORIZATION AND REPLACE THE ALTERNATOR, WHICH HE DID, AND MY CAR RUNS GREAT. THERE REMAINS, THOUGH, THE PROBLEM WITH THE HEADLIGHTS FLASHING. SINCE THE MECHANIC DID RESET THE COMPUTER, WE WERE STUMPED. THEN MY HUSBAND FOUND A SERVICE...

WHEN BRAKING, THE HEADLIGHTS PULSE. WHEN BRAKING, THE CAR RIDES BUMPY. THE STABILITY CONTROL LIGHT WILL COME ON AT RANDOM TIMES. THESE HAVE BEEN HAPPENING FOR THE PAST COUPLE YEARS, AND ALL MECHANICS HAVE SAID THAT EVERYTHING LOOKS FINE ON THE VEHICLE.