You want to control the +supply to the strips?
I would go an easier route - use 4 standard shift registers, with the outputs controlling current sink transistors, either N-channel MOSFETs or NPN transistors, and switch the - of the LED strips to ground. Control the segments that turn on from software, easy to do.
I'd go with something like this:
N-channel, logic level, low Rds, low input capacitance, low cost, capable of desired current
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Logic Level, Low Rds, N-channel. Can control from 0/+5 level, will not get hot, sink current to ground.