I have much experience with this. Getting a new cartridge will not help. The gold/silver cartridges have a battery inside them (which powers the in-game clock and also holds save files). Like all batteries, it dies after a few years.
Because gold/silver cartridges are no longer in development, buying a new one will only give you another dead battery. So, you must replace the battery. It's very simple. Any generic watch will contain the battery you need, and after that it's just a matter of replacing the battery (for which you'll need a screwdriver and a soldering iron). After that, you can start a new game and saving should work fine.
Would this happen to other Pokemon games?
It happened to my Gold, but I don't want it to happen to Blue aswell.