Check the wiring of the shift registers. Either the schematic is wrong, or you have connected them wrong.This is explanation and shows cascading for 2 shift registers:http://bildr.org/2011/02/74hc595/You have enough pins, so you don't need cascading.
I'm going to be using LED strips. I thought I saw current limiting resistors on the strip. If they have them on the strip would it be necessary to include them on this board?
Shift registers:Shift registers can be used in many ways, but is it useful ?Pushing 24 bits into the shift registers is very fast. I think there is no need to change it.
Led strip resistors:The led strip is for 12V. Often 3 leds in series are used, with a resistor, to make it work for 12V. With a led strip, you don't need extra resistors.
fadeTo get all the RGB colors, a PWM signal is used for every 'R', 'G' and 'B'.Since you use the shift registers, using PWM is a lot harder and not so smooth anymore. Did you know that ?
maximum currentI see you have the 12V at the led connector, and the ULN as driver.The ULN chips have a maximum current per output and a total maximum current.Can you check that ? The current depends on the length of the led strip.
Maybe late to the game, but I would just use qty 3 TPIC6B595, in place of the HC595 and ULN2803. 150mA current sink per output, rated to 50V. 3 LEDs/strip are generally 20mA, so 21 LED strips would be supported.There are other versions of TPIC6x595 that can sink more.I have boards on hand already that have arduino functionality and 3 TPIC6B595s already made if you want to buy a bare board, or an assembled one. Will post a pic when I get home. All thru hole, easy to assemble.
Pin 7s are floating, tie to +5 if not used.Are you sure you don't want to use Q7 of I.C. 2? Edit maybe you want "see image"
If you have an free pin on the controller, route it to the outside of a three pin male header.The other outside pin could be connected to +5V. The middle pin of the header connects to pins #7.Place a header jumper on the +5V or controller side as needed.In the controller position, you can send a clear.I'll bet you are getting comfortable with your schematic drawing program, that's the way you learn.