Uh, yeah. 350 watts is NOT enough for a GT240, with everything else in your system.
It really is amazing what you can get away with these days. I build work PCs for our employees here at work, using 300 watt power supplies, but we use integrated or very low power low profile video cards, and the systems only have one very silent 80mm exhaust fan (other than the power supply fan). You are likely not encountering an overheating problem now that we know that your PSU was 350 watts. The 550 watt should be fine, but it depends on brand. If you bought some off brand for $20 that "says" it's 550watt, then you're probably not doing much better than the 350 you had before.
Your problems sound exactly like a video card deprived of power. I just hope you didn't try to use it so much that you caused permanent damage to the card (which yes, can happen with low power)