Since you've cleaned the heatsinks, reseated all cards and RAM it's time to consider it needs some new parts.
A Pentium 3 is beyond it's expected lifespan so it may be best just to return it.
We recently bought a reformatted, used PC, a pentium 3 w/996 MHZ and 320 RAM. The computer was working fine until a couple of brownouts from storms. Now, when the computer heats up, it will shutdown automatically without warning, as if someone pulled the plug. We checked all the usual items, wires, cords, wall jacks and all are ok. We removed the cover and saw that the fans were working. Then we put a fan blowing on the computer as well. However, the problem persists. Any suggestions?