Well I just had to RMA my old 7850 for that exact problem. Since we basically have the same setup sounds like the card is bad.
I'm assuming that you have a A10-5700 processor which is like my A10-5800K.
One thing to check is with the 6670 you had Dual Graphics and would have your monitor hook to the built in ports. With a 7850 you must hook your monitor cables to the cards port.
So if you have tried the cables on the card with a black screen RMA the card.
Let me know.
AMD A10-5800K, XFX 7850 Ultra OC Edition 975MHz 1gb, 16GB RAM, Win 8.1 Pro 64 bit, MSI 7778 motherboard, 500W PSU, HP P7-1235, CCC 13.9 WHQL