Inspiron M5030 random restarts [HELP]

I have an old M5030 running Windows 7 and it would restart at random ranging from 3-5 minutes, whether I'm at the login screen or at the desktop screen. I booted safe mode and it does NOT restart. I could run the thing for at 25 minutes no problem. I assume it's a software issue or the CPU was not overworked enough for it to be overheated in safe mode. I then rechecked the fan and repasted the CPU. The fan was fairly clean with little dust. The laptop still runs pretty hot though. I install malwarebytes and ran a scan in safe mode, found two trojans, and removed them. It still restarts though. I forgot to mention, but there are no error messages or anything like that after restarting. It just boots up like normal.

I'm currently running diagnostics, so whenever that's done I'll post an update.

Update: There seems to be no error in the diagnostics test. Safe mode works fine. Safe mode with networking oddly enough restarts the laptop again.

Update 2: I've uninstalled the wireless adapter and the laptop was able to stay on for a good while, but the wireless adapter had issues. Installed the latest drivers and restarted. Wifi is fixed. This time the laptop started acting up again and restarted randomly again. I'm assuming it has something to do with connected to the internet, as that's when the problem arises.

 

Well, it does seem to be a software issue, so have you tried, you know, re-installing Windows? (or at least running off a Linux live CD or something)

 

Unfortunately I do not have anything like that right now, and I would prefer to have this issue solved without resorting to reinstalling the whole system. That would be my last option.

Update: I was able to fix using Kaspersky's TDSSKiller. So if anyone else has a problem like mine, try using this program first. If anyone wants to know more go HERE. Specifically the second page.