Has anyone else had this issue and managed to fix it?

Some things that should make it more stable that worked for my case :

- download the 4GB patch that will allow the game to use a bit more memory (a lot of the crashes is because the game is 32 bits but memory leak enough to go over the 2GB 32 bits limit, so at least with 4GB patch you can play for longer)
https://ntcore.com/4gb-patch/
put the "4gb_patch.exe" into C:\GOG Games\Dragon Age Origins\bin_ship\ then run it according to the instruction in the linked page.

- in the Video option of the game set the "Graphic Details" to the lowest possible.
Even if you have a good computer it does not matter, because the game will memory leak .
So it's better to leave as much GPU or RAM free as possible to play longer, and playing with "Graphic Details" on lowest free a lot of RAM, making the game more stable for a longer time. The game is great so it's worth it even on lower graphic details

- there's this bugfix mod that will fix some stuff (not sure if any are responsible of crash, but better have less bugs than more) :
https://www.nexusmods.com/dragonage/mods/4689/
Note : this is for english version, if you play DAO in another language, this will turn half the dialogues into english while the other half will stay in other language.
A fix for that is to use this custom build of this mod, available there :
https://github.com/humhue/DAOTools/releases/latest
The dazip file must be installed with
C:\GOG Games\Dragon Age Origins\bin_ship\DAUpdater.exe

Then once correctly installed, go to
C:\Users\YourUserName\Documents\Bioware\Dragon Age\AddIns\qwinn_fixpack_3\core\
and delete the folder named "audio" to avoid a few dialogue lines being forced to english.

- avoid any "high defintion" mod like the "Dragon Age Redesigned" , those will definitively make the game eating more GPU and RAM, leading it to crash more often because of that dreaded memory leak.

Like this, you should be able to play for a couple of hours without crashes.

Now if you have a modern GPU that drivers support Vulkan 1.3 , you may want to give a try to this :
https://www.nexusmods.com/dragonage/mods/6435
That is basically DXVK and so use Vulkan while lessening the RAM usage so you can play even longer

The 4GB patch was for me a solution. Thanks @Sanc for your post. I haven't had a single crash since then. I think GOG should definitely mention this on their site. In the past, I never needed it for the DAO disc-version (DVD Release) ... only for the GOG version suddenly.

It's unfortunate that there is such a need to patch the game yourself to make it more stable.
I remember reading that gog applied the patch on some game they're selling in the past, i wonder why it was not done for DAO too
By example for The Elder Scrolls Oblivion on gog, the 4gb patch that enable LAA (large adress aware) is already applied :
https://www.gog.com/forum/general/mod_spotlight_fallout_3_new_vegas_tes_iv_oblivion_415de/post25/?staff=yes

I agree with that. I don't understand a lot of what GOG is doing here. Among other things, TES IV: Oblivion was offered in German at the time, complete with full dubbing. And then at some point later the German version was removed (presumably because a DLC was only available in English - but removing a language version completely for that reason is questionable). But be that as it may, Dragon Age Origins is now running.