dontlistentome :
AMD no longer supports cards that old with new drivers. So a lot depends on the games you play, which you do not name.
Also, enabling vsync isn't a good idea unless your framerate is always 60fps or higher, assuming your monitor refresh rate is 60hz. Due to the way vsync works, having it enabled if/when your framerate drops under 60fps makes things worse not better. So if by 'solid 60fps' you mean it only drops into the 50s once in a while, disable vsync.
Your CPU may affect things as well, but since you didn't mention your CPU or the name of the games you're playing I can't say if this might be part of the problem.
Sorry for taking up your time, I believe I have figured out a solution to my problem.
First off, windows 10 machine, using an Athlon II x6 1100T, with 8GB 1333MHz DDR3 running in dual channel.
I played Overwatch, Fortnite, and Minecraft, all on the lowest settings possible.(with the exception of minecraft)
I used vsync to see as if it could help, but not so much at the time. Okay, here's what I found that works for me. I used the latest, beta release of AMD's Crimson Edition 16.2.1 legacy driver (2016.) Upon installing, I only installed the display driver, because I noticed that sometimes it would install the audio driver first and cause a lot of issues.
Audio driver before video driver seems to be a common problem with older hardware, I have seen a few cases of this on the forum. This helped a lot.
Finally, I downloaded the RadeonPro app
here. In RadonPro, I navigated over to the "Tweaks" tab, selected "Vsync Control" and chose Adaptive vsync. Forcing triple buffering seemed to help in some games that could benefit from it as well.