Transparent Background Issues

Those pixels look like they exist, maybe you cut this out of a background? Easy old school way to get rid of them:

Hold command/control and click on your layer to select the image

Go to the Select menu, modify, contract, use 1 or 2 pixels, inverse your selection (cmd/cnt + shift + i) and delete the pixels. Try saving again...

If your image Mode was changed to Lab Color where neutral maximum black is 0 (on a scale of 0=black, 100=white} and the  a. and b channels would also be zero, you would find that:

Patch 1 is not completely black and slightly blue

Patch 2 is a lighter black and also slightly blue

Patch 3 is a much lighter black and a greenish yellow

Patch 4 is the darkest black of the samples but not completely black and a yellow/green

To repeat: the criterion, I assume, is zero, zero, zero

It would help if we could see the layers that created the logo. I assume there were several.

I suggest that, if you have Illustrator, the logo would be more successful if it was prepared in that vector app. It produces a crisper image and its quality is unaffected by size change.

May be you are exporting to PNG-8 with white as matt colour. If so try PNG-24 with transparency.

I think your background isn't a transparent as it seems - select the space around the logo and hard delete it. Also as an extra tip - if you don't want that thin white outline around the eagle - use an inner stroke or inner glow that matches the dark shade of the logo - adjust it so it doesn't creep into the inner details.

Is this graphic extracted from white background? If yes then you need to work harder to remove ghost outline. There are a couple of ways or techniques including new Select and Mask workspace.