Holding a 100R or similar value resistor onto the power pins for a few seconds when the board is disconnected will completely discharge the capacitors which should reset the ADC. What sometimes happens is on the I2C addresses where one of the address pins is floating instead of being pulled high or low the ADC can get confused when you change the address as any internal capacitance on the address line could make it read as still being high or low.
You could try selecting addresses where the pins are pulled either high or low, so 0x68, 0x6A, 0x6C or 0x6E.
Do you have a second Raspberry Pi you could test the boards on just to make sure it isn't a fault with the I2C bus on the GPIO header?
It is unusual for both ADCs to stop responding, normally when a board stops working only one of the chips fails. When both fail to respond it usually means there is a problem with the I2C communication. You could try reflowing the solder on the small 6 pin mosfet U6 as a dry joint on that or the multiway resistor R1 would stop it from appearing on the I2C bus. Heat each pin on the mosfet in turn with a clean soldering iron tip and make sure you don't accidentally bridge the pins.
Have the ADC Differential Pis been used to measure voltages or is this the first time you have tried using the boards? If there was a voltage over 2.048V on any of the ADC inputs it could have damaged the ADCs which would cause them to act erratically or stop working completely.