The bad news is that a swollen battery is indeed a problem, the biggest one being that it can rupture and cause a substantial fire. As you've already seen it can also deform your laptop, and I've seen it damage trackpads and keyboards as well, necessitating replacement.
The good news is that on your model the battery is easily replaceable, requiring only a P5 and T5 screwdriver and about five minutes of your time. You can remove the battery now and continue to use your laptop on its AC power adapter until you can get a new battery for it.
I also have to ask: If you have "so many important files and everything" why don't you have a backup? Time Machine is built into the OS, is incredibly cheap to implement (requiring only a cheap USB drive for storage), and requires zero technical skill to set up.