Things may not be as bleak as they sound right now. I have lived with 5 bad eggs for several months in my pc and my game is still running, the graphics are unchanged, my Pokemon are the same, there was no game freeze. The eggs have not taken over and spread everywhere. The extent of the duplication was one extra egg.
Trying to handle the bad egg is more likely to cause trouble than just leaving it alone. The worst stories I have heard about bad eggs come from people who withdrew the egg, kept it in their party and perhaps even hatched it or tried to remove items from it.
What I would advise based on my personal experience is to leave the eggs where they are. You can't release them. Avoid viewing their summaries to be extra safe-- there are reports that opening the data can sometimes trigger the game to freeze, though this did not happen to me. Most importantly keep your Pokemon at a safe distance, I store mine a box or two away from bad eggs.
I would also recommend reaching out on the Rom Hack section too so you can get in touch with more players of Shiny Gold Sigma who might be going through the exact same thing and can offer you their expertise.
Good luck, I hope you are able to get back to enjoying your game.