Hi Jsnyd77,
Thank you for using the Apple Support Communities. Being able to set downtime limits is definitely important, and we'd like to offer assistance. As we understand it, you're able to set a schedule under Settings > Screen Time > Downtime, but the settings don't seem to be saved. Is that right?
To verify a few points, is "Scheduled" switched on there? Are you tapping "Turn On Downtime Until Schedule" at any point?
When you're setting the customized schedule, are you using Every Day, or Customize Days? Knowing the process you're using and which exact switches or features seem to be changing will help to narrow this down.
Also, are you using iCloud and Screen Time for Family?
The best steps will vary depending on the above, but if you haven't already, it might help to do a restart of the device in question and then try configuring it again: Restart your iPhone
Keep us posted. Take care.