Hi [USER=121352]soullifter[/USER] -
Welcome to PSG! Hopefully we can help you figure out what's causing the loss of sharpness.
The first thing I did was to take a look at your first in-line attachment (...the one titled, "Trump SC"). Unfortunately, it confused me because it appears that all of the emotes in it are 56x56 pixels, twice the 28x28 size that you are having problems with.
To try to reproduce the problem you are having, I took a large version (from the vector) of a crying emoji (about 1300 px square), added some text, stroked the text, added (like you) a slight drop shadow, shrunk the whole thing down to 56x56 (using "Bicubic Sharper"), and finally, for easy, direct comparison with your examples, inserted mine into your first attachment. To my eye, at 200% zoom in PS, the sharp edges in mine look just as good as the others in that panel.
[ATTACH]69043[/ATTACH]
BTW, just in case you don't aren't aware of some of the idiosyncrasies of some on-line forums like this, you should never rely on in-line images in the forum to do accurate comparisons, particularly, around sharp edges. This is because the in-forum mid-size previews are usually re-sampled and compressed dramatically to speed page loading. Instead, you should click on the image, expand it to its full, original dimensions, download it, and view it in a high quality image viewer like PS itself.
Back to your problem ... When I have to produce images like these, I never start the work at small image sizes (ie, low numbers of pixels in both directions). Instead, I always start with a much larger image than my final target size, get it looking good at the large size, and then, as the very last step, down-rez to the final (tiny) size. I would *never* do more than one re-sizing (ie, re-sampling) step, and I would never start with an image that's only a few times larger than the intended final size (ie, like your experiment with a 128 px image).
The best way to get us on the same page is if we both are working on the same image, so, attached below is the large version that I started with. My suggestion would be for you to download it, do whatever you usually do to down-size it, post it, and then, we can at least see if we get the same results, discuss methods, etc.
HTH,
Tom M
