Game crashing within minutes

Hi,

My system specs are :

AMD Phenom II X3 720 BE
NVIDIA GTS 250 512 Mb
4 GB Ram
Pretty cheap 400W power supply( Thought it has decent amperes on the 12V rails)
etc.

I have an extremely weird problem. A bit of background

I had an HDTV before which could support only 720p, and the graphics card worked flawlessly with all games at high/very high settings(including crysis etc.)

Now, ever since i got my Full HD LG TV, as soon as I begin playing any game at 1080p, the screen blacks out, stutters, etc. and the display loses the signal. Note that the computer is still running and if i press Alt F4 the display returns to the desktop in all its glory.

I am using EVGA Precision to track the graphics card temps, which seem normal - 45 under idle, and about 75-80 under load. I have also used Furmark and the same blackout happens followed by no signal for even the basic tests.

While i understand that a GTS250 cannot run games in Full HD properly, I expect the game to lag instead of crashing.

Now, my questions are as follows:

1. First issue i guessed was the power supply - i presume most of you will recommend a change. A doubt though - is the power requirement of the card "on demand" or should I have experienced instability during normal computer use.

2. At 720p do games like crysis stress the card fully at high settings or is the CPU involved?

3. The graphics card is able to hold full load for not more than 3 mins. Could the graphics card be faulty?