The symptoms you describe sound very much like a CPU chip overheating and causing the system to shut down until it cools off again. Then you say you believe the smell is coming from the CPU cooler area. I suspect there's a real problem with CPU cooling, and not with your PSU. Check these items.
1. Observe the CPU cooler fan carefully. Does it start up right away? Does it slow down quite quickly, then speed up until the system shuts down? If it does those things, then the fan itself is probably OK.
2. Check the CPU cooler fan bearings. With everything still shut down, spin the fan with your finger. It should spin easily and keep spinning for a little while, slowing down slowly. If it is stiff, or stops really fast, the bearings are bad and you need to replace the fan.
3. Check the fastening of the CPU cooler heatsink module over the CPU chip in its socket. Normally it is held in place by four screws through it and the mobo into a backing plate in the back of the mobo. Are all those screws there? Are they all tight? The heatsink should be in snug contact with the CPU chip, with no wiggling or tilting.
4. Carefully remove the FAN that cools your CPU from the rest of the cooling system. That is, leave the heatsink part of your CPU cooler attached to the mobo and CPU chip. Just remove the screws that fasten the fan to the heatsink. Now examine the heatsink and its fins. There should be completely open space between the fins. Often dust accumulates in those slots and reduces heat removal. I have even seen heatsinks so badly fouled that the entire space between fins is filled with packed dust and it can't cool anything! If there is dust accumulated in the fins, clean it out carefully, then re-mount the fan on top.
5. If you've made some changes and improvements above, re-assemble the system, leave the case open so you can see, and turn it on. Does it do the same as before, or is it better, or even fixed? If it's fixed, close up and be happy.
6. If your machine still is not working properly, you MAY need to replace the thermal paste between the CPU heatsink and the CPU chip top surface. This can be a bit difficult on three points. First, it can be hard to separate the CPU heatsink from the CPU chip if the old thermal paste is hardened. Secondly, the remains of the old paste need to be cleaned off carefully but thoroughly before you replace it. And lastly, the amount of new paste, and how to apply it, needs to be done exactly as the paste maker recommends. If you need to take this step, post back here for more detailed advice before starting.