I’ll look into the themal paste as well since that is an easy thing to replace.
Just wanted to mention, because it's often overlooked, check the thermal paste on your GPU as well. My wife's system was crashing constantly in games and it was driving us crazy trying to figure out why. Then, I pulled the GPU cooler and found the thermal paste was more "powder" than "paste". Cleaned the old paste off, applied new paste, and the system ran like a dream for another year or so before she upgraded the GPU.