Lines that are present on the screen, but not in screenshots, mean that the lines are coming after image processing. So the GPU and memory are clear, but sending that image to the monitor is being corrupted. That means: faulty output port on the card, faulty cable, faulty input on the monitor, faulty monitor. At least that means it's easy to narrow down.
Elimination method: Try a different monitor with a different cable from a different output port on the card, then track it back.
Tedious/diligent method: Try a different cable first, try a different output on the card second, try a different input on the monitor third, try a different monitor fourth. If you have a known-good monitor (with a known-good port) and a known-good cable, that helps to narrow it down quickly.
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