I have confirmed that when I push the button and turn the regulators on, they work as expected, and I can measure the 3.3V and 5V with a multimeter. They are also successfully powering the load, which is a microcontroller on the 3.3V rail and some LEDs on the 5V rail.
However, I've found that when I turn the power button off, I'm still seeing voltage on the output of the switching regulators. ~3.9V on the 5V rail and ~0.9V on the 3.3V rail. I know there might be some leakage through the components from the batteries to the rails, but this seems pretty high to me. It's high enough that some of the LEDs on the 5V rail are on, even after I hold down the power button to turn off the LT chip and PWR_EN drops to 0V.
Is it normal to see this much voltage drop from the battery to the switching regulator outputs, even when the EN pin is being pulled low? I've tried grounding the EN pin directly to make sure the LT chip wasn't stuck in a weird state.
What am I missing here? Why aren't my switching regulators fully shutting off? Isn't driving EN/!SHDN low enough?