I see that you are in Paris TX. Do you have that extremely expansive, black clay there? Do the power poles along the roads all lean over to the side?
There are some places where the dirt just moves so much that you have to go to extremes to deal with it. Six feet down for fence posts. You cannot use wire to tension your corners or H braces, it has to be something solid like pipe.
Every winter, the ground gets saturated by all the rain. Then it gets cold and the ground freezes. This is what destroys foundations. The movement of the soil is huge because it holds so much moisture in it just like an ice cube tray in the freezer and the ice cubes pop out of the tray. The frozen ground pushes the posts out of the ground, or moves them around inside the ground if the post isn't far enough below the freeze line to withstand this movement. Then if the Spring, when the soil thaws out, and the ground is nasty mud, everything starts to move. You rarely see it when the freeze does the damage, you just see it when the soil releases the hold it had on it as it melts.
Lots of people will not even use wood posts for fences because it moves so much. 3 inch steel pipe that is welded together, and so far into the ground that nothing can move it.
If you have nice red clay like I do here in Tyler, then you need to dig out around your posts by hand. Do not use the backhoe. That will just loose up more soil then you want to deal with. Get the clam shells out, buy or rent a small jackhammer. I use a SDS Max with a spade digging bit in it. You want to keep the sides of the hold in tact and solid. Then pour your concrete and use metal or wood for your bracing.