Everything else works, but my screen keeps showing weird stuff every now and then. There are no physical cracks, but there...

Sounds like it's an issue with your battery. You may need to replace it. You can find a new one online. Also, check out the guide for replacing the battery: Sony Vaio Fit 13A Battery Replacement Hope this helps!

I've found this suggestion previously and it worked well for me: What kind of adhesive for tablets and phones?

You may have to replace the battery. My laptop was having the same issue. Other chargers didn't work and the outlets were working. After I replaced the battery, it had no issues with charging. I hope this helps!

Is this the teardown you're looking for? Samsung Galaxy Note 10.1 Teardown

Vanessa, First, make sure the charger is completely plugged in. If it is, it may the outlet that isn't receiving power. You can test that by plugging something else into it. If that still isn't working, you may have to replace the battery. The same thing happened to my computer. Replacing the battery fixed the issue. I hope this helps!

Stephen, The vents to your computer's fan may be clogged. Try cleaning them. If that doesn't work, you can check out the troubleshooting page for the device here: Sony Vaio Fit 13A Troubleshooting Hope this helps!