How do CMOS switches behave when Vdd is grounded? What are the alternatives if they behave undesirably?
You could use the OMRON G3VM-21LR10 mosfet relay: -



They are very small but a bit pricey. You can get ones with 1 ohm on resistance and 1 pF off-capacitance making them quite good for audio switching. They even make versions of the device that are changeover thus featuring a normally closed contact but I think they are 15 ohm on resistance.

Read the data sheet for full details.