So here is the issue and where I need some experience. I spread the third load on the area in front of the shop and around to the basement garage. Graded it all with the box blade and compacted it with my bobcat (also compacted the dirt underneath with my bobcat the best I could). Looked great! I noticed immediately after it got some moisture in it that when driving on it cars just SINK IN. I mean bad. Like nearly getting stuck in it. I am feeling a bit defeated at this point but I really need to get it fixed before winter gets any closer and sets in.
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You are getting all sorts of advice, some I agree with, especially from Jaxs, others are very localized to where "they" live and not what I would do if I was where you live.
You said that you have a high clay content, which is what I was expecting from your pictures and your location. Soil that is mostly clay does "NOT" have a layer of organic materials, or topsoil. Do not try and remove topsoil when you do not have it, all you do is create a place for water to settle down into and sit. Kind of like digging a very shallow pond.
Since water likes to sit on top of clay, drainage is extremely important. That hill to the right of your picture looks like it's providing a lot of run off when it rains. You might also be getting a bit of a spring effect with moisture that soaks into the hill above, coming out where you cut into the side of the hill. Water goes down into the ground, then comes out where it can.
You are going to have to create proper drainage for this area. A big ditch is always best, but it requires the most work. Pipes work, but they tend to clog up over time.
How thick is the rock that you used? Is it road base rock with different sizes from a couple inches down to fines all mixed together? At a minimum, road base needs to be at least 4 inches thick. More is better, but if you go less, it will not interlock together and form a solid road that will shed water.
When you compacted it, did you add water to it at that time? If you compact it dry, it will appear hard, but when it rains, the water will soak into the rock and will make it soft. Just like you described. Once this water evaporates out of the gravel, and as you drive over it, and if it's thick enough, it will compact for you and become very hard.
If you do not have enough rock, no matter what you do, it will never lock together.
Fabric is for poor soils, wet areas and those in a hurry and not wanting to, or unable to prepare the soil. In most cases, it's not needed and just an extra expense.
To compact rock, you want as much weight on as small of footprint as you can get. I use my backhoe with a full yard of dirt in the front bucket, and drive over it with the front tires, over and over again.