Ditch those 74HC595 chips. Look at STP08DP05. It's also a shift register, but each line has a constant current (up to 100mA per line, 800mA for the whole chip) LED driver. You can daisy chain the shift registers by dumping the serial out of one into the serial in of the next, so you use 4 lines for your whole project. You also don't need resistors. I think there is a voltage overhead of 0.6V or something thereabout, so you ideally want your power supply to be just about 0.6 volts above your LED forward voltage - that way you aren't dissipating too much heat in the shift registers.
EDIT: If you need a through-hole package, look at TLC5916. Pretty much the same as the above STP08DP05, except it's in a DIP package and supports 960mA total ground pin current, more than you need.
DIP won't have the same power dissipation as TSSOP though, so again, be sure to run your voltage supply at the lowest level possible for your LED's to minimize dissipation in the chip (it's a linear device after all).