Older CRT monitors (and TV sets) generally did a pretty effective degauss if you turned the power off and then back on. It's possible that the mysterious recovery in the first instance was just because you did an (accidental?) off/on power cycle sometime during your fiddling with it.

For several old monitors I've had, there wasn't a degauss button, so that was the only method available for cleaning up the picture. If your monitor has a degauss, it's probably a "modern" CRT - maybe not even more than 20 years old. [friendly "dig," of course]

The cables generally used between computer and monitor usually are fairly well shielded, and with a single-cable connection cross-talk problems should be fairly rare, even if your "messy" turns into a rats' nest.

Should you decide to replace the monitor with a flat screen (a good idea, usually) the biggest problem you'll find is that the people who sell monitors now think that everyone should have a 72-inch monitor that is NOT MODESTLY PRICED and you may have trouble finding something that will fit in your room - much less on your desktop (or in your budget).

A significant advantage of a flat screen in my case is that it leaves space on the desktop for all the bottles of little pills I have to take daily so that I can keep them orderly for check-off as I take them. In her case it makes a comfortable space where the cat can sleep in front of the monitor - giving her mostly a cat's ass to look at that's much more interesting (IMO) than her Facebook. You'll need to assess what personal advantages a more modern screen would offer for your own use.

John