Try resizing your image in Photoshop first to the actual dimensions you will be using in your Illustrator files and re-link your image. I suspect the large dimensions are what is causing the wonkiness.
Here's a fully non-destructive way to create something similar using Photoshop. The beauty of this approach is that you can adjust the filters (or even remove them) if you don't like the result, without doing the whole edit again.
Right click the image layer, and choose Convert to Smart Object
Click Filter > Camera RAW filter.
Click on the fx tab in the ...
You can double-click on the picture and choose picture shape, in the middle of the picture tools tab, select Soft Edged Rectangle next to the picture styles and then use the Crop tool to crop it to your desired shape.
Click Crop > Crop to shape
Select one of the shapes
Illustrator is a vector imaging program. I think that RAW is not included as a compatible raster format for Illustrator and so I am not sure it can handle RAW files properly. I have no idea why the jpg version don't work, but it should work just fine if you convert the images in PSD file format.