Analog STDP switch using tactile button with state control
Basically you want a 16PDT (electronic) switch. Look up analog multiplexers, that's how they're usually implemented. Here's for example the functional diagram from the datasheet of a 4PDT one: 



You'll need use four of these.

You also need some additional memory circuitry to remember the state without power, like NOR flash, e.g. found in a microcontroler, if you want to completely avoid a mechanical way of remembering the settings.

And you can still find DIP packages for both analog multiplexers (e.g. BU4551 is the same type as the one above) and microcontrollers (e.g. ATMega328P aka arduino, or a smaller one like PIC12F617). You can find multiplexers with more (than four) channels on a chip, but usually not in DIP because they get large.