Photoshop: Retouching images to white; the white is dulled and you can see the retouching when I share the files

It sounds like a monitor calibration/ icc profile issue to me. Is your monitor calibrated? If not, a grey can easily appear white on your screen.

Are you talking about the background needing to be pure white?

Add a Levels adjustment layer and move the black slider all the way to the right (it will be on 253.) Now duplicate that layer.

Anything in your image that isn't pure 255 white will show up easily. The Dodge tool works great for cleaning up highlights, try setting it to around 3% with a hard edge.

Your display is probably too bright. Try turning down your brightness to 50-60%.