I think I have found the answer to this problem. I simply logged out of the launcher all together and logged back in and somehow that made the launcher sync up with the website. I am now able to change my skin on the launcher and it will say "Success! Your skin has been uploaded!"
So for me, logging out of the launcher and logging back in resolved this issue.
I've tried to upload a few skins that I got but none of them work. They all give me an "Unable to Upload Skin." error. I also tried to upload the skins from the website which says it worked but when i get in the game, I just have the Alex skin.
You're welcome! I dont know why it took me so long to figure that out. In my experience with computers I learned that "resetting" or logging out/in fixes most problems.