I'd start with an inverter, to take the single "master control" signal, whatever you want to call it, and create an inverse of that as well.
Then two identical output circuits with an instant-on and delayed-off function.
They could be eg. an OR gate with one input direct from the appropriate control signal and the other input via a resistor capacitor diode type delay from the first input.
(ie. capacitor on the second input, charged via a lowish value resistor and diode from the first input when the signal goes high, discharged via a higher value resistor when the first input goes low).
Two of those, one on each the signal in & out of the inverter.
At each input change, the new output becomes active instantly and the old output "holds on" for whatever time is set by the capacitor on the second OR gate input discharging.