In this case you should connect the cathode (short pin) of each LED to a common ground, and the anode (long pin) of each LED to its respective shift register output pin. Using the shift register to supply power like this is called sourcing current. Some shift registers can't source current, they can only do what is called sinking current. If you have one of those it means you will have to flip the direction of the LEDs, putting the anodes directly to power and the cathodes (ground pins) to the shift register outputs.
To sink more current, yes. But a tpic6c595 does the same with one chip.You still need to find a way to source more current for the anodes. Neither uln2803 nor tpic will do that for you. Transistors may be the answer.
Isn't the ULN2803 just a transistor array, or did I miss something?