My power adapter seems to be having trouble connecting with the power jack in the back of my laptop. The effect is really visible because my screen brightness changes everytime I move from being on adapter power to being on battery power. I've been working on looking for fixes for this.

Today, however, I just got home and my laptop was on stand-by, which is odd because I'm positive I powered it down before work, so it must have powered itself on and then went on stand-by after 20 minutes.

It's an old Dell laptop (Inspiron 8600), and I'm hoping to get a little more time out of it, but should I be concerned about safety if it's turning on by itself now?
Thoughts?