I was responding to your original post and trying to come up with a suggestion that would fit your design. Sinking posts would require some changes to that design. Since you said you were new to building, I wasn't sure how ready you were to look at design changes. There are countless numbers of different ways to accomplish what you are trying to do.
To be honest, I was not thinking about frost heave when I originally replied. After sinking foundations 4 feet deep in Kazahkstan to get below the frost line, I should have. My mistake. Something else I omitted. This structure really does not look all that heavy compared to its surface area in the coop section. I'd have concerns about the wind blowing it off the foundation if it is not secured.
The local conditions do matter as to what you do. Frost heave and erosion are two of the concerns, but you can also get movement depending on the soil type. One example I am very aware of is, if you build on reclaimed marsh land, the soil will shrink if it gets dry, swell if it gets wet, and kind of ooze out from under the foundation if it gets wet under the foundation, especially if it gets some type of cyclic loadings, say due to wind or heavy trucks going by. A friend in the New Orleans area had 22" of differential settlement in their house that had been built on marsh land on what turned out to be a floating foundation. Piling was the correct answer for that foundation. I'm not specifically suggesting that for yours, just giving an example.
The design you have is going to act as one structure. Many designs have the coop and run acting independently but not this one. How much a structure distorts due to deflection is a function of its stiffness. Since this does not have diagonal bracing, the run portion will not be very stiff which means it will distort easily. As long as it does not pull the run apart, that is probably not a big deal. The coop portion will be a lot stiffer which is important because windows and doors can stick. If the coop portion is not firmly fixed to the foundation, the distortion is a function of the stiffness of the construction (materials and construction techniques) and also a matter of time. Wood under stress will distort over tiime. If it were stiff enough, half the supports could just go away and it would not really matter, especially in the short term. Think of an empty cardboard box. If you support it under three corners, it will set there forever without distorting. But if you start filling it with dirt, at some point it can really deflect. Very few of our constructions are stiff enough that this relative distortion would not be a problem. I would not count on it in your case.
If your coop and run is attached to a foundation, then the stiffness of the foundation comes directly into play. If the supports move in relation to each other and your structure, then in addition to the gravity loads (weight) and environmental loads (wind), the relative movement of the supports can add a tremendous load and cause great distortion. Look at it as the stiffness of the entire earth versus the stiffness of your structure. The earth wins every time. So you need to assure the supports do not move relative to each other or your structure. There are different ways do this depending on your specific conditions. Posts in the ground act as piling and can be very effective. Spread footings, like gravel, contained sand, or concrete spread the load into the ground and work well. Some ground is stable enough that just putting the support on top of the ground will work. No one solution is the right solution for everyone. It depends on your local conditions. It's probably the engineer in me, but I have trouble understanding why so many people have trouble accepting or understanding that. Anyway, that is my problem, not yours.
I know I am overanalyzing it and probably boring you with this long discussion. In the end, you are looking for a foundation that does not move relative to your structure above. Many different ways to achieve that.
I don't know if any of this helps you at all. Hope some of it does. Good luck!!!