I'm a little confused by the first part of the reply. In the circuit I provided, one shift register is connected through it's data, clock, and latch pin, as well as ground and 5v. The second shift register is connected via the same clock, latch, ground, and 5v as the first but it's data pin is being cascaded. This configuration still suffers from the fact that the 7-seg displays draw too much current for the 595 to handle.
This might be a total noobie question, but how would one go about using one adapter to both feed the Arduino and the circuit? I realize there's a Vin pin on the Arduino but people caution against using it.
As to answer the second part of your reply, I would like to keep the 7-seg displays either entirely separate or in 3 pairs. Each pair of displays would have a header (months, days, hours) engraved above them in the MDF material. Hopefully this beautiful sketch below helps explain why:
Thank you for your time!