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when I try to save my game, I get a message telling me that my game entries are full and I need to replace one. Is there a way to delete older entries?
Go to the folder where the game is located, usually

C:\GOG Games\The Adventures of Willy Beamish

then you'll see some files named SCORES.0xx

You'll probably want to erase all but the most recently saved one (click the date modified tab to make sure that is the most recent save)

I erased all but SCORES.001 so I have no idea what will happen if you delete all but any SCORES number that isn't 001, but it freed up the space and I can make multiple saves again. Maybe back up your old saves just in case before erasing them though. (I did this with the talkie CD version)
It should be mentioned that the saves (Scores.0xx) files as mentioned above, can be used on both the Disk or CD version of this game :)

One just has to place them in the proper directories.