Hi, sschastain.
I'm glad you decided to reach out on Apple Support Communities. If I understand correctly, you attempted to change the Screen Time passcode on your child’s device, but it stayed set to the old passcode.
I understand the important of being able to manage your child’s device, and I’m happy to see how I can help.
First, you mentioned the steps you took to adjust the passcode. Be sure that you've followed these steps from Use Screen Time on your iPhone, iPad, or iPod touch.
"To change or turn off the passcode on your child's device, go to Settings > Screen Time, and tap [your child's name]. Then tap Change Screen Time Passcode or Turn Off Screen Time Passcode, and authenticate the change with Face ID, Touch ID, or your device passcode."
I want to make sure you changed your child’s passcode instead of your own. If you did, you might try again and this time restart both your device and theirs after.
Best Regards.