Hi Rufus, thanks for the detailed answer.
Heating up the chip causes the leakage current to increase by a lot. I guess this doesn't really prove that the boards themselves have leakage.
Could it be capacitance from the pads to VDD? I only have 10mil clearance from the pads to a VCC pour. If it was just board leakage, would the leakage current behave like what I observed, where it decreases like an RC circuit to some constant value? Here's my layout:
Putting a resistor in parallel would solve the problem... but it makes the PCB design much more complicated and would increase the cost a lot. There's 24 outputs per chip.
Do you think I should wash and soak the entire board after it's assembled? It's not that convenient for me to get purified water though... I'm sure tap water wouldn't make it any better.