I don't think that's your PC. I actually measured how much power my PC and my monitors consumed in a week and used that to calculate how much that would be for the year and compared to the total used energy for that year. My PC setup was only a small fraction of the yearly usage. The vast majority of your energy is gonna be consumed by things like fridges, ovens, heating, water pumps, etc.

Any halfways decent GPU driver and device firmware will put it into a low power state when it's idle.

Has the actual KwH increased on your bill? Power bills are like 50% fees so double check the fees aren't the thing catching you out.

I built a new machine pretty recently, also with an RX 7800XT GPU (factory overclocked). When sitting idle at the desktop, the system draws about the same amount of power as my old machine did with an RX 480. So I think trying to put the big GPU to sleep during desktop use might be barking up the wrong tree.

I suggest getting a power monitor, like a Kill-A-Watt, and taking measurements while you experiment. Here are some ideas to consider:

• Are you using multiple monitors? I have read that newer AMD GPUs sometimes draw more power than they should in this case. It might depend on the resolution and/or windowing system in use. (I don't remember if the reports I read were on Wayland or Xorg.) It almost certainly is a driver issue.
• Are you using nonstandard timings? Have you tried different refresh rates? https://community.amd.com/t5/graphics-cards/which-monitor-timing-parameter-allows-gpu-vram-frequency-to/td-p/318483
• Have you been playing games for hours every day, with no frame rate limit? The graphics card can draw considerably more power pushing polygons at 1440p@180Hz than it does at 90Hz, for example, and I don't think the wattage progression from idle to full load is linear.
• Are you using recent kernel and firmware versions?

That should be mostly the default. My secondary Vega 64 is reporting using only 3W which, on a laptop would be worth it but I doubt 3W affects your electricity. It's nothing compared to the overall power usage of the rest of the desktop, the monitors. Pretty sure even my fans use more.

The best way to address this would be to first take proper measurements. Maybe get a kill-a-watt and measure usage with and without the card installed to get the true usage at the wall. Also maybe get a baseline with as little hardware as possible. With that data you can calculate roughly how much it costs to run the PC and how much each component costs, and from there it's easier to decide if it's worth.

Just the electric bill being higher isn't a lot to go with. Could just be that it's getting cold, or hot. Little details can really throw expectations off. For example, mining crypto during the winter is technically cheaper than not for me because I have electric heat, so between 500W in a heating strip or 500W mining crypto, they both produce the same amount of heat in the room but one of them also made me a few cents as a byproduct. You have to consider that when optimizing for cost and not maximizing battery life on a laptop.

For this to work with a desktop PC, you would need to connect your display cable to your iGPU instead of your dGPU. The driver should take care of the rest. This might yield lower performance when using dGPU for processing (probably unnoticeable, depending on circumstances).