Analog Switch times for popless audio
Page 13 of the ADG1419 datasheet, figure 28 reads \$Q_{INJ} = C_{L} × ∆V_{OUT}\$. Charge injection is specified on page 4 as 13 pico-Coulombs, which isn't much, but must be enough to do what you are seeing. So this formula is saying that the amount of injected charge is the product of the output capacitance and voltage swing.

To reduce the effect of charge injection from this chip, decrease the output capacitance \$C_{L}\$ on the output pin 1 AU_OUT (perhaps add a unity-gain op-amp buffer right after this chip with a tiny input capacitance.) And if you can, decrease \$∆V_{OUT}\$ which means decreasing the 7.5v rail the chip is powered from.

Unlike a FET switch, this device most likely cannot be "slowed" to produce a noise-less switch. 

However, the duration of the glitch is around 4µs, and 4µs corresponds to a frequency of 250kHz. If your audio content only extends to 20kHz or so, placing a 20kHz low-pass filter after the switch may completely remove any glitches.

Another (brute-force) option could be to add a transistor which pulls the output strongly to the signal's reference voltage. Turn this "clamp" transistor on and flip the source switch simultaneously, then turn the clamp off 1µs later. The clamp will absorb the majority of the charge injection, and lasting only 1µs, the lack of output may not be noticeable.