Well this is a fine kettle of fish! I'm noticing artifacts on the screen now.
Before you go shopping for new GPUs, there are a few things that might help the artifact issue that I would try:
1. Replacing the HDD with a SSD - The HDDs run really hot in those models, and I suspect that when the HDD starts to go, it might be running hotter than usual. If you replace your HDD with a SDD, this might help the GPU issues.
2. Clean the Dust - If you never had your iMac opened before, that is 10 years of dust that is being stored in there. I use my air compressor with a blowgun attachment to clean out my electronics. A compressed gas duster would work too.
3. New Thermal Paste and Pads - That is 10 year old paste you are using, replacing this could cool things down a lot.
4. Bake your GPU in the Oven - Many people breath new life an old GPUs from the Late 2009 - Mid 2011 iMacs by baking the GPU in an oven. This worked for me.
5. Do them all - A combination of a few things from above might be able to fix your GPU issues. Personally, if I was taking the GPU out to put new paste on, I would just do the oven bake as it would only add a little bit of time and effort. While I had my iMac open, I would clean up the dust and replace the HDD with a SSD.