Cleanly cutting screws is easy when you have the right tool. Strangely enough you probably already own this tool.
To cut a screw you will need a pair of combination crimper & wirestrippers, which is probably already in your tool box. Simply thread the screw into the tool where indicated by the red arrow, and squeeze until the screw breaks. After removing the screw you may find a small burr which can be removed with sand paper.
To cut metric screws you will metric wire strippers (for metric wire), I bought mine on a trip to Europe at Praktiker Baumarkt. Unfortunately the brand I bought was not that good.
Kombizange zum Crimpen und Abisolieren
Gewinde abscherer von M2.6 bis M 5
I managed to find online nice pair of Knipex brand wire strippers online, however I doubt they ship to the US. Maybe eBay...