Moved WP site to a new host. Admin does not save info to the new server

Hello!

I moved my WP site to a new host. It looks great.
However, if I make any changes through Admin panel, none of these gets saved on the new server (but instead the changes get saved on the old server).

BUT, if I take another computer, open my website and log in as Admin, then whatever changes I make through Admin panel do get saved on the new server.

I need to make my computer work with the new server though. I restarted the computer, emptied all browser memory, restarted browsers, re-saved permalinks – nothing helps.

I would appreciate any advice here. Thank you!