I want to sign out of my iPhone but keep all data, does anyone know if Messages, Screen Time, Call history, etc. stays on the phone?
Yes, but you will want a backup regardless.
This sequence is part of “unsharing” an Apple Account: How to “unshare” an Apple ID - Apple Community
When you sign out you will be asked if you wish to keep data on the iPhone. I believe there will be a list with selections to remove or keep (I am not sure what the option actually says since I have never had to do this).
This support article appears to address your question --> What happens after you choose Sign Out But Don't Erase - Apple Support
Will data remain on iPhone after signing out?
