If AMD processor then the thermal paste they use themselves is perfectly OK and should not be changed. Check location and seating of processor and ensure you are using approved heatsink and fan. If you have bought boxed AMD processor with heatsink and fan included these are guaranteed for 3 yrs. There is very good detailed instructions on AMD web site regarding installation of AMD processors. I guess Intel have the same
I wonder if above posts re power supply might point to the problem. Depending on what hardware you have, especially graphics card, 300 watts is by no means over the top these days.
Can you try system with on board graphics or old minimal graphics card? Then it will be possible to check in BIOS if temps are OTT causing shut down
Check the fan connection to the motherboard is in the correct socket, as some turn off power if they do not detect a cpu fan. Also if you run some of the larger cpu fans from Zalman with speed controllers they can be too slow for the motherboard to detect, so disconnect the controller and modify the bios when it boots to stop shutdown.