1998 Chevy Cavalier 1998-Chevy Cavalier

HI,
I HAVE IS A 1998 CHEVY CAVALIER. I HAD THE CAR FOR THE PAST 10 YEARS IN FLORIDA. THE CAR WORKED FINE. NOW I RELOCATED TO UPSTATE NY, BEFORE I LEFT FLORIDA THE CAR HAD THE TUNE-UP AND CHECKED HER BEFORE GOING ON THE ROAD. I GOT HERE IN NY ON SEPT. 08. THE WINTER COLD HITS' AND HERE IT GOES; STARTING THE CAR IS OK AT FIRST, MAYBE 10 MINS. STILL DRIVING DOWN THE ROAD THE ENGINE LITE GOES ON. I BROUGHT THE CAR TO SEARS. THE TECH RECOMMENDED A WINTER BATTERY. A DIAGNOSTIC WAS DONE BUT NO CODES CAME ON. I STILL HAVE THE SAME PROBLEM, AS THE CAR STARTS AND ABOUT SEVERAL MINS. LATER GOES OFF. IS THERE SOMETHING I'M NOT DOING.