How do I get my game ( icon back ) to work again for offline gaming. I used to be able to play offline but now I have to log into GOG anytime I want to play now !?

Go into your gog folder and make a new shortcut it worked for me it might work for you

Thats what sucks is there is no .exe file to make new icon !?

Which game?

Sometimes games will be under a /bin folder. Some that use DosBox will run the exe from the dosbox folder supplied, and link to the .conf files.

More often than not i do see a 'launcher' shortcut that you can copy since it will already be hard coded for the location the exe file is at.

Assuming you are running Windows, have task manager running when you launch the game and check out where the executable is lurking.
Alternatively if you already know what the executable filename is, run a basic file search on your HDD.

look for a .exe file that launches the game

If all of the suggestions above fail...you can always just pick & use a custom icon, independent from the game in question. Kinda annoying IK, but it's the easiest way to solve your problem. ;)