I know the obvious signs of a blown computer with dead LEDs, blown caps, and clearly burned/fried slots.

I have an old Core 2 Quad system that i've been running some .BAT game servers off of for a while now.
Basically the CPU runs hot all the time now and it stutters quite a bit doing any basic tasks like browsing.

One of the DIM slots died on me last year and Im pretty sure this thing is on its death bed.
I've tried everything I could think of to keep the CPU cool but not sure what to do anymore

Move on?

Yes? Just Craiglist a Dell PC for like $60 instead of continuing to deal with the headache imo

Probably just overheating; when was the last time the TIM was changed? I have been running core 2 machines 24/7 for years.

Hello sir, I will be assisting you today.
What model of motherboard are you using?
Are you overclocking your CPU/RAM/GPU/PSU?
How many DIMM slots does that motherboard have?
What operating system are you running?
How many fans does your system use?
Thanks. Looking forward to hearing from you so we can diagnose your issue.