You mentioned ATI graphics, so if this is an older video card, you may need to test the workaround posted in the sticky thread for current Classic issues:
Game crashes in Fullscreen mode on older AMD graphics cards
There is a known issue where certain supported graphics cards may crash while in fullscreen mode. To fix the problem, you can launch the game in windowed mode:
Open the Blizzard Battle.net Desktop App and select the WoW tab
Click Options and select Game Settings from the dropdown menu
In Game Settings, activate the option Additional command line arguments. Type in the following:
-windowed
Retest the game.