"I plan on using 16ga speaker wire to power the positive and red leads of each strip. "
So these are strips with +, and R, G, B contacts, and you are only using R?
Then TPIC6B595 would be the way to go. 6 LEDs per segment would use ~ 40mA/segment. Each output can sink 150mA.
I offer a board that supports up to 12 of the chips and has a 328P for controlling them. Plug on an FTDI Basic or equivalent for programming it.
http://www.crossroadsfencing.com/BobuinoRev17/
Here the board is driving an 8-digit test board I have with 3 and 6 LED segments.
