Get a new power supply. After you work with it using Sidran32s excellent advice for checking.
The reasons I am going with that advice, it sounds like your PSU is not "up to date" and as power supplies age, the capacitors age. If it was a cheap psu to begin with , it could be weak, and dirty. The new card might put you in a good position of upgrading the PSU to a better quality , and newer version, even one that is more efficient , more quiet and better quality.
If you need an excuse to update the PSU , and the psu is aged or junkey. If the psu does not have the multiple GPU wires and High amp 12V rails, they put on the new ones. it might be a good time to do that when putting in an expencive GPU, and trying to make sure everything works well.
Then you still have the First problem :-) which should be easier to deal with when you have one less item in there to worry/wonder about.
Side notes: - - - - - - - - - - - - -
Some of the Motherboard connections are NOT compatable with PCI-E connections because of odd things they do different with the wiring, like extra ground on 8pin PCI-e. So checking your polarity by at least looking at the color coding of the wires , and checking manuels and any polarity + - signs on the cards (where the power connects).
If you use a Modular PSU the wires are NOT always the same between manufactures.
These 2 things have had some serious consequences for some people , who learned while they burned. When making the changes and adaptions, take extra time to check things, before the plug goes into the wall.
Remember a PSU is not "off" when the computer is in the off state, you must turn off the power supply switch, or remove the plug from the wall (or both).
I have had situations where a video card (and the psu) were fine and the card just would not Init right away. It would just not work, and the computer would not boot, and persistance with it paid off. When it finnaly booted it took 5 minutes for some strange reason. Other users have reported that happening, I still do not know what to attribute it to.