Dell laptops are notorious for cracked mother boards causing the kind of problems you described. They are heavy machines and when you pick them up with one hand, they flex. Over time, that leads to trouble. This is exactly why Steve Jobs loves his unibody Macs so much.
However, it's impossible to tell if that's your problem without poking around. My suggestion would be to take Bubba up on his offer. $59 will give you peace of mind relative to the chess club and if repairs are possible and cost-effective, the $59 applies. That's a reasonable deal.
Be sure to screen and check his P411. And make sure he uses latex gloves.