Use the switch to control analog switches. The switch then just needs to be SPST with a pullup resistor. This will be the control signal to the analog switches. One switch line can easily drive many CMOS inputs, so fanning out to six won't be a problem.
Analog switches are sometimes called "bilateral switch" because, unlike "digital muxes", they work just as well in both directions -- much like a relay.
Sometimes I use an analog switch to connect the output of one chip to either one of two "input" lines (with pull-up or pull-down resistors to set the "other" line to the appropriate state), much like a digital demultiplexer.
Sometimes I use an analog switch to connect the input of one chip to either one of two "output" lines (no resistors are needed for this case), much like a digital mux.
Sometimes I use an analog switch to connect a bidirectional line like the I2C data line.
Analog switches act much like relays, except
(a) they can't handle much current -- some handle 25 mA, others can handle only 10 mA.
(b) they can handle only a limited voltage range -- some handle 0 to +15 V, others handle only 0 to +5 V.
(c) they are lower cost, can switch far more rapidly and more often, and require much less power than relays.
(This is a minor tweak of the answer from Olin Lathrop).
Some parts I would consider:
- 74HC4066 quad SPST bilateral analog switch
- 4053 triple SPDT analog switch
- 74HC4353 triple SPDT analog switch
- 4052 dual SP4P analog switch
- 74HC4852 dual SP4P analog switch
- 14551 quad SPDT analog switch
You might be able to do this with two chips, and perhaps some pull-up or pull-down resistors.