What do you mean by 'stuck in manual mode'? Are you saying that the automatic transmission isn't shifting?
Traction Control works by detecting wheelspin and limiting engine power and/or applying the brakes to stop the wheelspin. I believe that you'll only see a light if you've turned it off or if it's broken. Ordinarly, there's no indication that it's working.
Get yourself onto some really slippery stuff, come to a complete stop, then nail the throttle. If the Traction Control is working, you will notice the car cut the throttle to quell the wheelspin even if you have the pedal floored. Keep the pedal floored. If the wheels spin away like, TC is not working. If wheelspin is more or less under control and the car begins to move forward, TC is doing its job.
Traction control makes it EASIER to keep the car under control in slippery conditions, but your car is still subject to the LAWS OF PHYSICS. So be careful, and think of TC as merely a traction aid -- not a miracle device.
That said, TC won't help you nearly as much as good snow tires. And without good snow tires, TC is gonna be next to useless.
TC makes the most of the mechanical grip that you have. If you have no mechanical grip at all (because, for example, you have shitty tires or you're on sheet ice) then there's not much the TC can do.
Think of traction control as a translator. It takes your hamfisted commands and translates them into smooth, gentle commands. If you're in conditions where even a smooth, gentle driver would have difficulty, then the TC will have difficulty, too.
These cars are still rear-wheel drive, and require care in slippery conditions. Get yourself some good snow tires, put a couple of sandbags in the trunk for extra back-end snow traction, and drive carefully.
Without good tires, no amount of caution or electronic trickery will help. It's all about tires.
And remember that, unlike summer tires, winter tires become useless with 50% of the tread remaining. With summer tires, you can wear the tread right down to nothing and still be okay (except in rain). With snow tires, once the first 50% of tread is gone they're ready for replacement. They may not look it, but if they've got 15,000 miles on them, they're no longer suitable for snow or ice.