I lived in a trailer as a kid in northern Maine. Year-round. In the winter we made a point of piling up snow against the trailer for insulation value, turning the trailer into a sort of igloo.
If you shovel snow away from the trailer, you can't necessarily get it all without risking damage to the trailer's skin from nicking it with the snow shovel. So unless you have to go inside the trailer, and unless you have to turn on the heat inside, I'd recommend leaving well enough alone until it starts to thaw, and then using a soft-bristle broom or brush to whisk the snow away.
Oh, and do not turn on the heat in the trailer to melt the snow faster; that's an invitation to the very leaks you're worried about. Let the snow melt from the outside in, not the inside out.
Engineering: Finding complex solutions to simple problems you didn't even know you had.