How Do I Change a Background Without Cutting Off 'Borderline' Pixels

[Taylor-eOS]

I searched for this everywhere, but can't seem to find an answer, even though it has to be a common problem.

I have an object surrounded by a background, in this case the white. But the transition between the background and the object isn't entirely clean. As you can see some of the pixels close to the border are a little darker.
If I change the background or select the object, it will either cut the transition pixels off and make the result seem cut out, or have this thin gray border around the object.

So is there a way to take the slightly darker white-ish pixels from the white background and transfer them to the new background, but not as white pixels but what's darker than white onto the new background?

I don't want to cut all the white inside the object, this is just about the cut-out.



Thanks