Title. Stuttering is very bad even in the practice range. Background apps don’t affect it as I have tried it with all apps turned off and it still didn’t affect performance. Restarting the computer gets rid of the problem. This happens in other games, but the most noticeable example is Overwatch.
Thanks in advance! 
Since this issue is not specific to Overwatch, I don’t know how much support will happen in your thread. But my best suggestion is to look at the guide to help maximize performance for gaming: GUIDE: Maximizing System Performance for Overwatch (PC)
Other than that, there must be something running in the background, that is causing the issue. I’d look at the task manager when the games are open, or even in the Windows Event Viewer to see if there are any critical errors.
What are your PC specs?
RX 580; 8GB RAM; i5 7600k (not overclocked);