Saturn L series. After winter set in my oil is brown and foamy. There is also smoke and the vehicle shakes violently when I try to run it. In what ways might the motor not be completely fucked.
Not very many I'm afraid. The foaming can be a result of water condensation in the system, or combustion gasses, which is reinforced by the shaking and smoke(white smoke, I presume). You either have a bad head gasket likely caused by age combined with ambient temperature, or the cooling system was mostly water and froze/expanded inside the engine block, causing all sorts of hell.
Luckily for you, those engines are rather easy to change and they are a dime'a'dozen. Check
www.car-part.com for a local scrap yard near you that has one in stock. You can filter by price and distance. Check your local craigslist or a local facebook group for a guy like me that does backyard work but knows what he's doing. All in, you could probably stick a new engine in there for less than $1000.
Bit off-topic, but I gotta say, I'm seeing more adverse affects this year from the cold than just about any year prior, and we're not even out of December yet. Strange.