Stay wide in the corners around the opponent cars. That way they'll pass themselves when they park instead of crashing into you and unsettling your car.
Obviously a good tune should help a lot. Try not to hit the curbs too hard unless you're sure the car is settled, and hit them going as straight as you can. If the car is still being too wild on the gas for you, try to be very careful about using second gear and don't bother with first. Don't be afraid to short-shift if you sense wheelspin coming. If your top speed is right you can get around Cote d'Azur decently enough to win without using second at all if necessary.
Remember, you don't need to squeeze every ounce of performance out of the car to catch the AI, if you remain patient and keep the car pointed in the right direction you'll catch them eventually over 20 laps(and you'll probably find it gets easier and easier to post faster lap times as you continue to drive without over-stressing yourself). Then dont rush to create a pass or try to stay glued to their gearbox, just stay out of their way until they slam on the brakes mid-corner(you usually don't have to wait long).
