Hmmmm, I don't know what's wrong with me this week. I'll add one now. Do you think that's what could be causing the instability in the data that I am reading?
Yep. The instantaneous current draw of a write can cause a heck of a sag in the power rail...at the chip...and cause the wrong data to be written, if it's written at all.
A smattering of .1uF caps across various powers/grounds will do no harm.
And the wires themselves are another story. If you've got a foot of wire for each connection, well, you're just inviting problems.
As far as your '595 set up...I'd be putting some LEDs on those output pins and making sure the pins are doing what you think they're doing vs. what you want them to do.
eg. LEDs on the data pin, write all ones to the pin, all the LEDs light up. Write a 10101010, every other LED lights up, and so on.
Same thing goes for reading. As mentioned before, hard wire the input pins to a specific combination, eg. 10101010, and see what you read back. And try various combinations until you're satisfied that you're '595 and inputs are working correctly.
In fact, tie the '595 outputs directly to the inputs (thru a resistor of course).
If you write something out, you should read the same thing back.
Get that SRAM (OTPROM, whatever) out of there for now.
Break it down. Build it back up.