Can you elaborate on frying the shift registers?
well you can:-
Right now I've been able to get away with no resistors between the shift register outputs and the LEDs
This means that the only thing limiting the current is the output impedance of the drivers. That means they are supplying too much current. You don't need resistors because you are killing your shift register outputs.
Read this about LEDs and the voltage / current relationship. http://www.thebox.myzen.co.uk/Tutorial/LEDs.html
Also how will changing the transistor affect the current?
Because the on resistance will be much lower and that allows more current to flow.
However, I measured the current at the output of one shift register pin (i.e. one LED) and it was like 1.5mA while lit.
I find that very hard to believe. Meters don't tell the truth when subject to any sort of oscillation or chopping.
any ideas what series to go with
Try the IRLB8478