Both those cards have 6-pin and/or 8-pin connectors on them to which you must connect the appropriate PCIe power cables from the PSU.
The cards will not work on slot-power alone.
Have you connected the aforementioned PCIe power cables ?
PSU is obviously working okay if it runs a 1060 with no problems. And both PSUs should be okay to run either of those Radeon cards.
That points to the possibility that the Radeon cards may be faulty. Can you test them on another PC ?