Is there a demux IC that allows a default signal (not just digital 0) to go to the non enabled outputs?
Use a set of 2-to-1 MUXES.

Most micro-controllers have two or more PWM generators. 
Thus you can use one to 'move' a servo and the other to 'position' a servo.

To switch between the two you use a 2-to-1 mux per servo where the inputs are connected to the two PWM signals. 

Now if you control each mux select signal separately you can switch between the two PWM settings. Your program should make sure that only one select is 'active' at a time to meet your requirements.