I have a similar problem with an external VGA display (HP LE2201w on a MacBookAir M1 with a brand-less USB-c to VGA converter)
I have not tested all programs that have a 'save' or 'save as' dialogue but I had the problem with NeoOffice (Intel) and Firefox (Universal) (both most recent versions).
After reading this discussion, I started testing what triggered the bug in relation to the Display preferences.
It turns out that if I move the Menu bar to the external display, the bug shows up. If I keep the Menu bar on the built-in display the problem does not show up, even if the application window is showing on the external display.
Further test results : If I start with the menu bar on the external display and have the application window open (the bug shows) and then move the menu bar to the built-in display, the bug remains. If I start with the menu bar on the built-in display with the application window op (the bug does not show) and then move the menu bar to the external display, the bug does show.
So for me, the menu bar needs to be on the built-in display before opening the application.
It's not pretty nor handy to have the menu bar on the built-in display which I use as the secondary display, but at least I can continue using the applications.