In order to do fast on/off switching of the outputs from the shift register to make sure that no 2 segments are on at the same time, would this require using PWM and is it possible to time the signals so that the peaks of the outputs never overlap but are close enough that it appears they are on at the same time?
OK, so we have adequately dealt with the original topic of this discussion and
given that it is only an exercise now and not something you wanted to do for real (because if you
do want to drive those 7-segment displays with more than 10 mA per segment group, you
do need to go and get some TPIC6B595s; no other approach is worth considering).
So ...
If
for some other purpose entirely, you wish to ensure that no two outputs of the shift register overlap, you do not need to do anything at all apart from shifting in a pattern which does not contain more than one bit active. Nothing happens until you activate the "latch" function on the shift register, at which point all the data you have shifted into it is conveyed to the outputs
simultaneously. You are confusing yourself even further by referring to "PWM" which has nothing to do with this problem.
But please do not continue with the theme that switching on 74HC595 outputs alternately will
in any way permit you to apply more average current to your LEDs. The chip is
not designed to provide more than the current specified, either per pin or total, and you will be particularly disappointed if you actually attempt to draw more than 10 mA at most from any output as in doing so, the available voltage will fall by at least a volt.
even before any damage occurs to the chip and it will simply fail to light the LEDs to the brightness you might expect.