Fundamentally, unless you want to continue doing this every year (which I suppose is an option), youhave to move the coop onto something that is not going to heave/thaw unevenly. Either another patch of ground that doesn't do that; or sink something (foundation, posts, whatever) deep enough into the ground that it will go well below your frostline and not be 'grabbed' by frost in the upper part, and set the coop on that.
The only other thing I can think of is, special case, IF your problem is simply that the coop is quite heavy and the south side ground is thawing earlier than the North side (because of direct sun on the S side ground vs shade on the N side ground), then you might be able to even it out by putting down heavy insulation in a wide swath around the S side AFTER THE GROUND HAS DEEPLY FROZEN and before it starts to melt. This would probably mean, when your snow starts to melt enough that the ground is almost but not quite exposed. If different thaw rates of the two sides is your problem (and I really don't know if this is the case, only you can guess that) then putting down, say, a 12" layer of shavings the entire length of the S wall and extending maybe 5-8' out from the wall in all directions will keep that ground frozen a lot longer. When the N side ground starts to soften then you can remove the insulating shavings layer.
BUt in most cases, what you describe is really only solvable by getting the coop onto something that basically won't move. (I don't suppose you have any granite shelf on your property, or other large rock outcroppings?)
Good luck, have fun,
Pat