a 350W PSU doesn't mean it is a 350W PSU.
f it a cheap one or the one that comes with the case it can get to 350W output, but it cannot stay there, it probably can't get to 300 and stay there.
What you have done if the wires have melted is tried to put too much current through the PSU down the wire and it has started to create too much heat.
If you smelt burning though, it probably is not the wires, it will be the unit itself.
To explain what it is i will tell you how the temperature protection works. If the PSU gets too hot there is a little component in there that if it gets too hot blows up, goes pop but inside the PSU keeping the extra current contained, it breaks the circuit making the fuse in the PSU get too much current and overheat breaking, cutting off power to the system instantly because it gets so hot so quick. Your PSU is then completely broken but your system is fine.
I would say deffinately get a new, better PSU, only maybe a 450-500W one, but one that isn't really cheap