Darkstone's old save system doesn't work well in a world of Cloud Saves. They save the PC name right into the files so saves and characters won't be visible when you boot the game on a different PC, even though they've synced to the save folder. If you try to create characters on the new PC, you'll wipe the ones created on your old PC. This is how to move your saves and characters safely to another PC.

IMPORTANT:

First, DO NOT CREATE A CHARACTER ON THE NEW PC yet. Cloud save will overwrite your characters.lpl file and your old one will be lost when your old PC syncs with the newer file. If you do lose it, and don't have a backup, you can play old saves but can't select the characters for a new game. This means no new game plus on a higher difficulty. I learned this the hard way.

Second, BACK UP YOUR SAVES AND CHARACTERS ON THE OLD PC.
The files are stored in in your GOG install directory wherever you chose, under Darkstone\Save.
Eg C:\ProgramsFIles\GOG.com\Darkstone\Save
You need to backup the "characters" and the "games" folders into another location to keep them safe from sync. This also means if you aren't comfortable you have a back in case of hex editing mistakes.

Now we can begin.
Download HxD from their official site. This is a free, safe, hex editor that we will use to directly edit the files. Don't worry, it's easy.
Once you've backed up your old files, and they're somewhere safe from being overwritten by a sync, go ahead and create a new character on the new PC. Start a game, save immediately, and quit.

Transfer Characters:
Open HxD. Click Open, and choose Darkstone\Save\characters\characters.lpl
The first line will have your new PC name in it, with the last two character repeated (eg NEWDESKTOP.O.P).
Click Open again, and browse to your backup. Open the original characters.lpl file.
The first line will have your old PC name in it, with the last two character repeated (eg OLDDESKTOP.O.P).
Highlight the first line of your new file. Right click, and choose Copy.
Highlight the same line in the old file and right click. Choose Paste Write (NOT Insert).
This will replace the old PC name with your new PC name.
Go to File, Save As, and save over the new file at Darkstone\Save\characters\characters.lpl
Open Darkstone and click New Game. Your old characters will be there, and the new one you created will be gone.

Transfer Save Files:
Open HxD. Click Open, go to Darkstone\Save\games , and pick the newest save.
The first line will have your new PC name in it, with the last two character repeated (eg NEWDESKTOP.O.P).
Click Open again, and browse to your backup. Pick the save you want to transfer. You'll have to go off the date modified timestamp.
The first line will have your old PC name in it, with the last two character repeated (eg OLDDESKTOP.O.P).
Highlight the first line of your newest save file. Right click, and choose Copy.
Highlight the same line in the older save file and right click. Choose Paste Write (NOT Insert).
This will replace the old PC name with your new PC name.
Go to File, Save As. Save into Darkstone\Save\games.
Open Darkstone and click Load Game. The save you modified will now be visible on your new PC. It will load and play even if you haven't transferred the characters.
If you try to view the save file on your old PC after a sync, it won't be visible from the load screen.

I was having an excellent run with a warrior and wizard, finding all the right spellbooks at all the right levels and so on. Then I got a new PC, and I couldn't load it. I fiddled a bit between them before realising I had to edit the files. Unfortunately I didn't backup my original characters.lpl before learning that the cloud sync had wiped it. I've transferred my latest save, so I can play on with them, and I'll probably win with them, but they'll be forever trapped in Novice difficulty so I may as well start all over again. All my characters, essentially gone. Hopefully this spares someone the same pains.