Daisy chain 74HC595 causes issues

I am appending all my replies in one post because the forum is stopping me from making too many replies as a new user.

@ dlloyd
I think 7805 is currently not shutting down because I have got same results with 7.5 V on supply. I am using a heat sink already and the current draw into 7805 is less than 200mA because the top 50mA are going directly into the display via the main supply (the same 7.5V in this case).

I think your suggested problem was appearing when there was no heatsink and 7805 was outputting approx 4V.

@TomGeorge and @ Paul_B:

The heatsink was removed momentarily when I thought I should replace it with a new 7805. But, after verification that 7805 is okay, I am put the original one with heat sink back on.

MarkT:
I tried putting on the 10uF caps on the input and output but it did not help my cause.

@ PaulRB
Previously suggested using the 3.3V to power up 74595. So, in order to achieve that I cut the Vin to shift registers. And, even before I could hook it up to the 3.3V from ESP, I realized that it produced the correct results. I don't know where the SRs are getting the power from now. May be through the LATCH and CLK or may be through the ULN connected next to it. However, it getting it from somewhere and working fine. The voltage at Vcc of SRs is 2.3V.

Now, I just need to understand why this is happening and if this is a sustainable approach?

Looking forward to your comments.

Usman,