Weird sagging sensation on the right side below the rib cage when laying flat on my back. Feels like an organ (liver, kidney) is resting against skin?

Get a medical eval. This could be a variety of things. I recommend making an appointment with your primary care to have a good physical exam and taking it from there.
Certainly not that. There are a dozen or so layers of tissue between your organs and your skin, so it's certainly not that. It could be something as simple as a strained muscle, or it could be an early (or very mild) case of shingles, or some other disturbance of a sensory nerve on that side. Let your doctor sort it out.