The short answer
A washer that shakes violently in winter is usually dealing with cold, not a broken part. The suspension dampers stiffen when the room drops below about 45F, so a load that would ride fine in July starts hammering the cabinet.
The three things to check before you spend a dime: the load balance, all four feet, and the room temperature.
Most of the time it's one of those. When it isn't, you're looking at worn shocks or rods, and those are still a manageable fix on many machines.
What's actually causing it
Cold is the amplifier. It rarely breaks a washer on its own, but it takes small problems you'd never notice in summer and makes them loud.
Start with the load. Towels, jeans, and bath mats soak up water and clump to one side of the drum. In a cold room the dampers can't smooth that out, and the machine rocks hard at 800 to 1200 RPM.
Next, the feet. Concrete floors in garages and basements shift a hair with temperature, and a foot that was snug in fall can lift slightly by January. One loose corner is all it takes.
If those check out and it still shakes, look at the suspension. Top-loaders ride on suspension rods with friction dampers; front-loaders use two to four shock absorbers. Both wear out around 8 to 12 years, and cold exposes the weak ones first.
Two less common culprits: shipping bolts left in a recently installed front-loader, and a loose counterweight or cracked spider arm on an older machine. The counterweight and spider issues usually come with a knock or grind, not just a wobble.