Hey there, and welcome to Apple Support Communities, across64!
From our understanding, the Do Not Disturb feature is remaining on outside of the scheduled time. We’d be happy to help out!
In some cases, Do Not Disturb can be enabled without being on in Settings when certain conditions are met. For example, if you’re driving, the iPhone will sense that and enable the feature. You can read more about this under “What to expect when Do Not Disturb While Driving is on” at: How to use Do Not Disturb While Driving - Apple Support
In addition, the article above can help “Turn it on and off in Control Center” and if you’d prefer to have this enabled manually you can “Customize features for Do Not Disturb While Driving”.
If this does not occur while driving, let us know more about the situation this occurs in. For example, does it occur only in the morning or at another specific time of the day? Do you see the moon icon on the menu bar at the top of your screen?
Kindest regards!