The card is 8.2" and the motherboard width is 9.5" which means you'd have a good 1.3" to work with. I looked at the picture and those SATA ports look like they take up less than 1.3" so I think you will be fine lengthwise.
Even if you aren't, the cooler of the card doesn't extend very far and the PCB itself is very thin. If it were to extend that far (which it won't), it would only block the top SATA port. You'll be fine. I can see a 5850 with it's own cooler (the kind where it extends all the way to the end of the PCB) covering 3 of them.
The 4670 would be too much of a downgrade for consideration if you are looking in the 5770 range. What do you need the card for?