Each of R,G,B needs its own driver, 20mA for each. Individually. Otherwise, all 3 are on.
If you wirewrap the board, it will take very little time to wire the 8 signals across the chip for a couple of chips:
Output Drive (mA) -24/24
(vs 74HC595 Output Drive (mA) -4/4)
Connect straight across the 574:
2:19
3:18
4:17
5:16
6:15
7:14
8:13
9:12