How to switch multiple signal lines with a single mechanical switch
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.