Most decent systems use less than 350W actual power if you check them with
a power meter. But you shouldn't run a power supply over about 85% of its
maximum power/current output for any of its rails.
So a 350W PSU should be used to more than about 300W load. Depends on your
CPU, how many hard drives, fans, optical drives, etc. you have.
If you go easy on the overclock, keep the case/PSU WELL ventillated so
it's not getting hot, and it's a quality PSU, you could probably borderline run ANY of these
mentioned cards and squeak by without the system crashing due to bad power.
On the other hand I wouldn't be surprised if you did high overclocking and had a
really hot room / case and the PSU gets old/dusty that it'd flake out or fail soon.
Look out for one of these $9.99 or $19 after rebate deals on a 450W or higher PSU
that you see all the time. I'd guess even a junky quality "Ultra" "500" or "600" would
be more capable than one that's only rated to 350 even though it's a better brand/quality.
Don't change the PSU right away if your system passes stability tests
with the current PSU, but keep an eye out for a great deal on a replacement at
some point.