Safari is logging out of all websites I visit, every time I relaunch

Open Safari with a Non-Private Window

Are you opening Safari with a new private window? If so then cookies are being removed, and you are always being required to enter all this once Safari is closed.

So, change it so it opens with A New Window.

Go Here: Safari Menu > Preferences > General > Safari opens with pane> Select: "A new window"

Are you running any anti-virus programs?

Try setting up another admin user account to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply        Pastebin

Etrecheck – System Information

You are welcome. Glad it worked!