Multiplexing 2 times 4 CC RGB LED's
For idividual control of all 24 emissions (8x 3-color LEDs), you won't likely accomplish anything less complex than 14 transistors/switches (2 rows &amp; 12 columns).

With that said, however, you could accomplish that control with 7 (or 5) 'bits' of output from your shift-register (would require a few more transistors, though). This would use 2 bits-color select, 1 bit row-select, then 4 bits 4-channel LED on/off control (or 2-bits to control one @ a time). This would take 6 cycles x 7 bits (or 24 cycles x 5 bits) to 'fire' all possible combinations, but would work with your 8-bit max width on the shift register.

I would recommend using gates, rather than independent transistors, if you can "sneak them in" past the "no other ICs" rule.