I had forgotten my screen time password. When I reset it with the forgot my password option. After doing so it still will not allow me in with the new password. What else can I do to change the password so that it works?

If you have forgotten your Screen Time passcode. There are two options.

If you are unable to reset the Screen Time passcode from this-> link then, you will need a computer for option 2. If you don't have one, borrow one from a friend.

Option 1

Use this link to reset your passcode If you forgot your Screen Time passcode - Apple Support

Option 2

Forgot? Click below to reset…

Click->If you forgot the passcode for your iPhone, iPad, or iPod touch, or your ...

If you have chosen option 2 then regarding your data, If…

1. You have an iCloud or iTunes backup, I am sorry there is no use of a backup because if you restore from your backup your Screen Time Passcode will be back. 
2. You have synced your data to iCloud, then what you will get back is what you have synced to iCloud and uploaded to iCloud Drive; the information that is synced and stored in iCloud are, Contacts, Calendars, Bookmarks, Mail, Notes, Voice Memos, shared photos, iCloud Photos*, Health data, call history, and files you store in iCloud Drive. (*-If you had turned on iCloud Photos, your content is automatically stored in iCloud.)
3. You have imported your Photos & Videos to Mac/PC then they are safely residing there
4. You have uploaded some of it Cloud Drives using the Files app (iCloud Drive, Google Drive, OneDrive, DropBox, Flickr, box, AmazonDrive, Adobe Creative Cloud etc)
5. If you have not imported nor you have synced or uploaded your data to cloud drives prior to forgetting the Screen Time passcode then I am sorry, all your data is lost.