Red horizontal lines and dots all over my monitor

Try pressing gently on the bezel near where the lines are. If they change or flicker when you press, it's a loose connector inside, not a dead panel. Still a repair job, but it tells the shop what to look at. Don't take it apart yourself if it's under warranty.

Since it follows the monitor to another PC, open the monitor's own on-screen menu. If the lines are drawn over the menu too, it's the panel or the board behind it, nothing on the PC side. Couple of things worth trying first: unplug it from the wall for a few minutes, do a factory reset from the menu, and turn overdrive / response time down to off. Overdrive set too high causes artifacts, and they show up worst on dark screens like you describe. If none of that changes it, check the warranty. Out of warranty a panel fault usually costs more to fix than the monitor is worth.