You should be able to change your Apple ID to the iCloud address associated with your Apple ID. If you made a new iCloud email address, that is a separate Apple ID. Log into your Apple ID account and use the iCloud email address that is listed there.
Apple ID - Manage
Note the restrictions in the article.
Apple ID Name Change.
What to do after you change your Apple ID or password - Apple Support I suggest doing this before you change your Apple ID or password.
Based on what I have read, I suggest you leave your Apple ID email address as a 3rd party email address because you have more flexibility if you need to change the email address. For example, if the Apple ID got compromised, you can change it to another address. I also have read of many people who were locked out of their accounts when they forgot their password. Since the recovery email address is the @iCloud email address, if you forget your password, you also can't access your email to get emails from Apple to recover your account. You can set a rescue email address, but for some people, this hasn't worked.
If you want to use a the new iCloud ID, when changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion.
How to switch Apple IDs on your iPhone or iPad.
How to transfer all files from one iCloud account to another.
What to do after you change your Apple ID or password - Apple Support.
To connect with Family Sharing, have the organizer invite the new ID to join Family Sharing.
