I bought an Xbox 360 E brand new a few years ago; it worked fine until a couple months ago. Now, when I turn it on,...

I’ve had this problem once; it turned out to be lightly corroded contacts on the memory card itself. If you haven’t already, blow out the card, as well as the slot in the console, like you would an old NES cartridge. Another thing you should do is rub the card’s contacts with a rubbing alcohol-soaked q-tip. I imagine another possibility, if you’ve only tested either slot with the same card, is that the card has become corrupt either from age or from losing connection [either from dirty contacts or being nudged] while saving game data.