Using NPN Darlington to switch on PNP
That seems like a lot of complexity. If you want to do it as an exercise, or a homework, that is fine. I'd do something simpler.

Firstly, P-Channel MOSFETs would be an easy way to drive the common Anodes.

The P-MOSFETs will need their gates to pull-up to the chosen LED supply voltage 6.3V to turn off.

AFAICT the MSP430G2335 MCU doesn't have 6.3V (or higher) tolerant pins. So the MCU pins will need to be isolated from the 6.3V. 

Use a couple of NPN transistors to interface the MCU pins with the P-MOSFETs. The P-MOSFET gates will be pulled up with a 1k or bigger resistor (use same value as base resistors), and pulled down by the NPN transistor. To limit base current from the MCU pin, put a 2k or larger resistor between the MCU pin the NPN transistor base.

Use a 'proper' constant current LED display driver for each LED cathode, instead of transistors and current limiting resistors. 

For example TI TLC5916 which has a shift register for data load, and 8 constant current drivers. Brightness is set using a single resistor for all 8 LEDs. It will be less wiring, and fewer components even than using a darlington transistor array and resistors. 

There are lots of other devices, and other manufacturers make suitable parts too, for example Maxim.

That's it: 


2 x P-Channel MOSFETs,
2 x NPN transistors (almost anything) to pull down the P-MOSFET gates,
4 x 2k+ resistors,
1 x 8-channel constant current LED display (TI or Maxim),
1 resistor to program LED brightness pin of 8-channel LED display driver