Burning Smell after being stuck in snow

As was said, you probably just toasted your tires.

However, I really think it would be wise if you checked your transmission fluid.
Unfortunately, wifey’s method of gaining traction could have led to damage to the transmission.
If the fluid is any color other than red, or if the fluid has a “burnt” odor, I suggest that you drive it immediately to an independent (non-chain) transmission shop for examination.