Hello,
One test I like to perform when I'm not having a half-bridge operate properly is to remove the MOSFETs and test the gate drive without the half-bridge in place. Is this possible and easy for you to perform in your system?
When you say output low, are you putting all gates low at the same time? If so, I would expect the half-bridge voltage to float to half of the bus voltage. This would not explain why you are measuring 6V on the outputs of each gate driver channel.
Are you measuring with a floating multimeter, or an oscilloscope? If an earth-grounded scope probe is placed on the half-bridge voltage, shorts can be formed.
RSchnell