Cameras struggle with scenes that have high dynamic ranges (e.g. shadows, or in this case dark fur/hair and a very bright sky).
Having said that, I don't see a problem in your corrected image. It allows you to see more of the detail in the fur.
Depending upon the mode of your camera, you look at exposure compensation. Decrease expsosure compensation a bit (e.g. - 1/3 stop or even - 2/3 stop). It's definitely counter-intuitive, but that will raise the overall exposure of this scene (and the really dark blacks will be lighter; allowing you to see more of the detail).
For more power during editing, I strongly recommend capturing your images in RAW. Then, with software such as Adobe Lightroom, or Canon's free DPP, you can increase shadow detail and/or raise the black level a bit. That will isolate your changes to the blacks/shadows.
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Ricky
Camera: EOS R5 II, RF 50mm f/1.2L, RF 135mm f/1.8L
Lighting: Profoto Lights & Modifiers
