Electronic – Any IC similar to ULN2003A with latch or enable/disable

It can work without a latch providing the unstable period while the data is shifted to all the drivers is small - you probably wouldn't notice the very quick flickering of the displays. Maybe if you could tell us the clock frequency this would help but, if it's running at 1MHz and there are 4 severn-seg displays then that's 28\$\mu s\$ for the update - not a long time at all and maybe the clock is even faster.

Item C on your first drawing holds the answer yet I don't recognize it - probably a GAL device.

ground connections

Check the ground connections on the shift registers -- that's the first thing I would check. If I were you, I would add another dedicated ground wire between the ground pin of each shift register and the ground pin of your 7805 (or whatever voltage regulator you are using) -- probably overkill, but how else are you going to rule out "bad ground connection" ? The "dim LED" symptoms you describe are consistent with a ground wire to a shift register getting completely disconnected.

Many CMOS chips -- such as shift registers -- often seem to partially work after we accidentally disconnect the GND pin or the VCC pin or both, leading to hours of frustration when we assume that -- because it seems to be working -- surely the power pins are correctly connected, and we hunt fruitlessly elsewhere for the problem. Guess how I know this?

Powering a chip through the input pins -- rather than the VCC and GND pins -- has been called a "dirty trick" and "abuse".

p.s.: I think word clocks are cool. I wish you great success.