How can I find the current draw of an SN74HC595 when OE is disabled?
Maximum is 8uA at room temperature, 80uA at up to 85°C. But those are very loose specifications. The typical Icc is about 27nA with a 5V supply at 25°C. As a rule of thumb, it will approximately double for every 10°C above room temperature. 



That's assuming all the inputs are near the rails and static. If inputs are not near the rails the current can go up considerably due to conduction of the input transistors, and similarly if they are switching. Cpd is specified as 400pF (typical) and there is an explanation of how to use that here. 

If the input strays too far from the rails, then shoot-through current can greatly increase the tiny leakage current broadly similar to this effect: 

(from this document):