Great, that's good info. So the first thing to point out, and someone else please correct me if I am wrong on this, but playing a game at a refresh rate higher than the display's refresh rate and not using V-Sync (vertical sync) will cause display tearing. As the monitor is displaying the image the game will have changed the image. What you end up with is tearing where one image stops and the next (newer) image begins. That is why games add V-Sync. It may not be very noticeable at times, but that is inevitable result.
Aside from that, you should be able to get a game to run at a higher refresh rate than the monitor's fixed rate. In this case, your monitor is capable of 60 frames a second, and without V-Sync the games may run well higher than that. Without knowing whether Adrenalin identified the games you list, by clicking Gaming on the lower left you should see a list of games identified. For each game there are several settings to take a look at. They are Wait for Vertical Refresh, Radeon FreeSync, Frame Rate Target Control, and Chill (FPS). For troubleshooting I would set Chill to Off, adjust the slider for Frame Rate Target Control to whatever value you want from the game, Wait for Vertical Sync to off, and Radeon FreeSync to Off. Within each of the games there should be a place to analyze your graphics display or similar terminology and let the game analyze the hardware. There may also be a place in the game where you can specify a frame rate.
In Adrenalin I would also look at Display, located on the bottom just right of center (third in from the right). The Display section deals with specifics about your monitor. There is also a Radeon FreeSync category here.
