im personally have zero issues with the game, and i play windows (aka win9x/me) disc version. (at some circumstances i have mouse lag issue, unplug one headset and plug another one solve the issue).
there is no such bug inside the game. and been doshox only got version shouldn't have any compatibility issues. idk why this happens. most likely some app interfere with the dosbox window and broke it. some overlays.
i have few solutions or bare workarounds actually. Ctrl+S make Quick save (and ctrl+q make quit). or try Alt+Enter to windowed and then again to Full-screen maximized the dosbox window (standard Windows hotkeys for make windowed/maximized).
And at next launch from GOGDOSBOXlauncher exe file - and not desktop link (replace it with launcher link from OS Start Menu) - try change some resolution and render settings in trying to prevent this. Start as admin to change settings (if its in Program Files it is mandatory, otherwise mayb not).
Try to close maximum possible apps from taskbar and sys tray, except very few very needed ones to you. apps that make some message window or overlays - like messengers like Discord, or Steam - is first candidate's to be closed (and not minimized to system tray which the can do on closing, use exit function).