When you installed the new CPU did you also install new thermal paste and cooler?
Also, there is practically little, or no, difference in power draw between your old CPU and new one. You would not have needed a new PSU simply due to making that change. Based on the CPU only, your old PSU would have had plenty of capacity with no need for a new one.
Changes related to overclocking, the addition of a large number of additional drives or fans, or a different higher draw graphics card would be the only or at least main things that would require a new power supply of larger capacity. Also, sometimes "bigger" does not mean "more". There are 600w units that probably cannot sustain the same output as a good 450w unit of high quality. What brand and model of PSU are your old and new units?
What is your motherboard model? Most likely you have tried to install a CPU that either doesn't work with the motherboard you have, needs a bios update in order to work or does not support the type of memory you have. Some AMD AM3 processors used DDR2 memory and I don't believe there were any or at least not many motherboards that supported the use of the FX processors with older DDR2 memory.
What is the motherboard model you are trying to use this FX 6300 with?
Most likely your problem is because you are trying to use an AM3+ processor in an AM3 socket that either doesn't support it or requires a bios update to do so.
Have you tried going back to the old CPU? If so, and it worked before but doesn't now, then the use or attempted use of the FX processor may have fried the board. I would need to know the motherboard model number in order to know if the FX chip is even remotely usable in that board.