Sorry, Modelworks is wrong on two significant points.
1. By connecting the prong on the adapter to the screw properly, you do NOT connect it to the Neutral side of the power supply line. And the Neutral side never was a substitute for a true Ground. It is true that most house supply systems in North America and many other places use a "Grounded Neutral" system and have done for many decades. However, it is vital to recognize that the Neutral line is ONLY at Ground potential at the breaker (or fuse) box where it is connected to a true Ground source. Anywhere else in the house the Neutral line is NOT at true zero potential (compared to Ground) because of its non-zero resistance and the current flowing in the circuit. For this reason NO house wiring system ever uses the Neutral line as a Ground substitute - unless, of course, some amateur has mis-wired the circuit. You either have a true Ground lead that carries NO current under normal operations, or you have NO Ground lead.
2. The screw in the middle of the cover plate connects to the metal frame of the duplex receptacle mounted in the device box. That frame is completely isolated from both the Hot and Neutral connections to the blade slots. (In the current wiring system with a third bare Ground lead and a third prong hole in the receptacle, the frame is connected to the Ground wire and U-shaped hole.) But by being mounted in the box with screws, it is connected to the box. HOWEVER, in older houses with only 2-wire cabling those boxes are not connected to anything electrically - they are merely mounted mechanically in the wall. So in those systems the box, receptacle frame and screw are all floating completely independent of all electrical connections. They are NOT Ground, Neutral, or Hot. Thus attaching the screw in those cases does absolutely nothing. Now, in SOME systems there has been an attempt to update by installing a third true Ground lead to each box in the wall, or by re-wiring using BX cable - the kind with a spiral-wound metal cover that is connected to the breaker box itself, and THAT IS connected to Ground. So in those cases the box MAY be Grounded, and then the screw does establish a Ground connection to the U-shaped hole in the adapter. The trouble is, you do not know about those possible updates without doing a good investigation to verify the presence of a high-quality (that is, extremely low resistance) Ground connection.
For OP's original problem, the best suggestion I can offer is to plug the computer, the monitor, and any other connected devices into ONE common power source - like, a power bar - so that all devices have exactly the same Hot and Neutral supply lines. Plugging into different wall outlets may NOT achieve that, and result in "Ground Loops" that are really circuit elements designed to operate at the same voltage levels and references, but not actually doing that because of unbalanced supply issues.