Ok so quite old but not very old. Ngl it doesn’t seem very likely that the motherboard is causing you graphics output issues. unless the PCI-E slot is faulty. Can you put the graphics card in any other slots? Either silver slot should work (although the top one will be the fastest).
Dumb suggestion, but given the board, you probably don’t have onboard graphics supplied by the CPI; you’re not trying to plug the monitors into the vertical sockets are you? I have seen several people do this while ignoring their actual graphics card’s outputs 
You’ve said you tried different cables; did you try different types of cable? E.g. if you’re using displayport for both, can you put one on HDMI instead?
Also can you connect both monitors to any other computer? Can you take them to work, to a friend’s house, something? Just to eliminate that it’s not one of them sending a bad signal back to the PC
