74HC595 +TLC5940 multiplexing 128 leds -> not enough power
If you were running the LEDs in series then yes, you would need a higher voltage.  You aren't though - otherwise they'd all be on or all be off.  You are running them in parallel.

From what I understand of your description you have the one output of the '595 going into the anode of the all the LEDs of one column.  The cathodes of these LEDs then go into separate inputs of the TLC for PWM.

And, as you say, the '595 can supply 35mA of current per output.  That is enough for to light one LED reliably.

You will need to supply 6 times that current for 6 LEDs.

The simplest way would be to use a single transistor and resistor per column.  For example, the '595 output connects to the base of an PNP transistor through a 1KΩ (for example) resistor.  The emitter connects to Vcc, and the collector connects to the LEDs in the same way the output of the '595 used to.  When the '595 sets an output low it turns on the transistor which then allows the current to flow from Vcc to the LEDs.




I don't know what the TLC5940 does in the way of current limiting - I haven't shown any current limiting resistors that may be required for the LEDs if the TLC5940 doesn't do that for you.


You can't use the TLC with, for example. a ULN2803 as they are both current sinks.  You need something which can be a current source, which a transistor can be.