You'll be fine.
Back in the 60s and 70s when I skied, nobody, but NOBODY, had an SUV; they didn't exist. Every once in a while you'd see some nutjob in a Jeep, with his plastic windows flapping in the breeze, and it's true that every once in a while the winch on the front would be helpful to someone in a ditch. But everybody else made do with regular ol' cars. And not modern balanced front-wheel-drive cars like your Corolla. In fact, the best car in the snow in those days was the lowly VW bug, which had the engine over the drive wheels just like you do. We made it to the mountains many times with our crappy Fords and Mercurys.
I'd even go so far as to argue that a sedan in the hands of someone who is patient and paying full attention to the road is much better off than a hulking SUV with a typical oblivious nitwit at the wheel. Four wheel drive is nice, but SUVs still handle and brake like trucks. I see SUVs stuck in the snow all over town everytime it snows, even here in the lowlands.