When I have this game fullscreen the edges of the game go past the edges of my screen. Is there a way to adjust the resolution of the game to prevent this? Is this purely a problem with my monitor (Westinghouse L2410NM) that I can't fix through the game?

Do you know the max resolution of your monitor? You might need to play around with the dosboxGK.conf file. Try a different scaling option. I don't have a test file here at work but I think you might want to start with normal1x
(If you're wondering, this version of the game runs at 640x480, natively)

Can you not set such things anymore directly at the monitor? It was possible in CRT times (well, the last days of my CRT sadly have come - in a few days it's replaced by a TFT or how you call that).
I can set fullscreen zoom, picture geometry etc directly on my monitor. If you can't I don't know a solution.