I am glad that you will upgrade your power supply.
In terms of the video card fitting into the case, the only sure way is to measure the space.
If you can get inside one of the cases, measure the distance behind the card to see if there is enough for the longer 5970. I think that you will need an inch or more.
Or search online for the dimensions of the case and look at the back of the case to the back of the drive compartment.