Confusing power problem. Please help.

Okay, so I have no idea what's going on with my computer. Tried searching the internet for hours for a solution or any information on this, and came up with nothing.

So a few weeks ago, I had to get a new power cord for my MacBook Air, as the old one was worn out and would no longer charge the computer. The new cord worked for a couple of weeks, but then things started to get a little weird. My computer would shut down when I moved it while plugged in – this never happens if I move it while on battery power, only when it's charging. And now the cord hardly ever actually charges the mac when it's plugged in, and when it does, the computer shuts down almost immediately. I have no way of knowing whether this is the battery or cord, and the nearest Apple store is a good four or five hour drive from where I live.

If it helps at all, it's a 2013 13-inch MacBook Air running OS X El Capitan. I've had it for about three years or so, but haven't had the battery serviced or replaced yet.