Controlling 13 outputs with a mcp23017 or 74hc595 and two uln2003?
The 150mA/120mA are absolute maximum numbers and you should not get anywhere near those levels if you want reliability. The output voltage is only guaranteed at 3mA source/8mA sink. 

If the relatively high output voltage of the UL2003A Darlington outputs (maybe 0.7V typically @20mA) is not a problem for you, that's probably your best choice to buffer the expander outputs. Very cheap and robust part and 7 outputs in one small package.