If you have a data backup you can move it to your new MacBook, otherwise you could buy a replacement battery to continue using your 2013/wipe the data and sell it.
Also don’t want to ditch it for no reason. It might not be the battery. Really need to figure out how to get the back of and take a a look at them. Having seen photos, it’s quite obvious when they are swollen.
How do I get service for my Mac?
To get service for your Mac, you can make a reservation at an Apple Store or an Apple Authorized Service Provider. Make sure you know your Apple ID and password before your appointment. Or, if you have a Mac notebook, contact us and we'll send you a box you can use to ship it to an Apple Repair Center. We'll return your repaired product to you as quickly as possible. Depending on where you get service, you might be able to check the status of your repair online.
Hardware service may no longer be available for older Macs. Contact your local provider to inquire about available service options.
Do you backup to the cloud? If so try doing one before anything else. Or backup to an external drive.
Do a search for Apple repair in your area. Many are ‘authorised’ by Apple.
Don’t plug it again unless absolutely necessary.
Normally happen to tinker with things (any MacBook owners would do well to unscrew the back cover and clean out the dust every now and then) but I took my MacBook pro to an Apple store and a replacement battery cost CHF200, parts + installation.
...then comes the interesting bit, they don't have a formal product recall/product replacement policy but they fitted a new keyboard and case as well as the ballooning keyboard had bent them. This was free, all part of the CHF200 battery fee. They were sort of whispering about this second part rather than admitting anything had gone wrong.
The Apple store was CHF200. But I first tried a local agent and they wanted a lot more, seems the don't want to handle the warranty issues in the same way.
Anyway, laptop is like new now with the fresh battery and keyboard.
Having examined it today there is no other evidence of swelling apart from the lid not closing. The lid is not in contact with any part of the base section when it “pops up” 1 cm, so it’s not being restricted from closing by contact with the keyboard etc. It lies flat on table, keyboard and track pad are fine.
I’ve ordered the necessary screwdriver to take the back of and see how batteries look. If I wasn’t 4 hours away from the nearest Apple store I would definitely drop by and get it checked by them. If battery looks swollen, hoping the local Steg can replace it for me.
Apple recently had to agree to supply parts for their machines to third parties so you should be able to get a "official" new one.
Whatever you do, don't go for a cheap alternative, like the power blocks it is a bad idea, blow up the motherboard and you really can throw your MBP away.
https://www.ifixit.com/Store/Parts
These people have been around for at least a decade, sell authentic kit as far as I can tell. They have MBP battery for 89 Euros, although be careful with this, it say 2009-2012 so maybe not the one you want.
Here a video to help you too, note it shows you how to check if your MBP needs a new battery, you can actually check it in the systems menu.
My MacBook system info says my battery condition is normal (despite being 9 years old!)
The inside of my Mac was absolutely filthy though