Sounds like alignment or balance issues, though i'm not sure how you being stuck in some snow caused it. Perhaps i'm just sleepy and misreading your post, but I'm reading it as such:
1. Driving along all is well, but it's getting too snowy out for the tires
2. Too much snow, time to pull over and get towed
3. On truck, all is well in terms of rigging it up
4. Dig it out the next day, drive it onto dry land a few feet away.
5. Pronounced vibration upon driving
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