Here is a bit of hilarity: I had installed Divinity 2 but not played it yet, so I just started to play that instead, and it will not let me save games either. I tried a few other random GOG titles that I have installed, and they worked fine. I quit.
Oh dear, that's not good :( The last thing I can suggest is to contact GOG support; http://support.gog.com/hc/en-us
and hopefully they'll have the answer.
Good luck and hope you get it fixed.
Well, sunovabeyoch. They got back to me with a list of suggestions, and one of them was to check antivirus software to make sure that the programs are on the exception/trusted list. So I checked my Bitdefender logs, and sure enough it listed itself as having blocked both Avernum 5 & Divinity 2. So I whitelisted them, and bang, problem solved. The weird thing is, I have used Bitdefender for years and never had that happen before, so it did not occur to me that it may be causing the problem now. However, I did just renew my subscription, and it downloaded a new version, so it must be something that Bitdefender changed. Why, I have no idea. Preventing any newly installed software -- that the user has chosen to install, and clicked all appropriate acknowledgments -- from writing anything AFTER the program itself has already been "written" to the computer? But that was it.
A whole helluva lot of work that I just put into adjusting Administrator & program/file privileges, and it was all for nothing. D'oh. Thanks for your help!