I do know that perhaps it is not the right way to use these switches.

Why do you think you need a make before break switch?

 

OK so I've found this,

DG411, DG412, DG413 Monolithic Quad SPST, CMOS Analog Switches



It states this is the summary but I'm unsure as to how you would do this.



The four switches are bilateral, equally matched for AC or bidirectional signals. The ON-resistance variation with analog signals is quite low over a ±15V analog input range. The switches in the DG411 and DG412 are identical, differing only in the polarity of the selection logic. Two of the switches in the DG413 (#2 and #3) use the logic of the DG211 and DG411 (i.e., a logic “0” turns the switch ON) and the other two switches use DG212 and DG412 positive logic. This permits independent control of turn-on and turn-off times for SPDT configurations, permitting “break-before-make” or “make-before-break” operation with a minimum of external logic.

EDIT I also found this TI Analog switch guide and the TS5A3154 I mentioned earlier was the only one that was MBB which seemed weird. I have no idea why they don't seem to do them much anymore.

 

If you don't have to switch extremely fast, look the OPTOMOS series of products. http://www.ixysic.com/Products/SSRFormA.htm Nice chips in general. Ton and Toff are different, so you could have some some advantage in what your trying to do.

I like LittleDiode. Delivery to the US is a bit slow, but it is tracked.

 

Please find attached, sorry if it is a bit rough, I do know that perhaps it is not the right way to use these switches.

Another way to prevent a significant transient is to put a switch across the resistor at the capacitor output and momentarily turn the switch on when making the input source change.
That way, any transient voltage change will be shunted to ground.