Please provide your hardware specifications and if possible the model of the PSU. It very much sounds like the symptoms a faulty PSU. I'd advice to do one of two things, in order to establish that it is indeed the PSU which is the culprit, before you spend money on a new one.
1) Grab a digital-multimeter and test the power-unit. You can find numerous reputable guides. This way is very quick and reliable in determining if the PSU can or cannot deliver the necessary power.
2) Borrow an already working PSU and swap it in your system.