As Michael stated, use pull-downs at the outputs of 74HC595. 2k2 resistors should work.

Spend a port line of micro (PORTx,y) to control the /OE of 74HC595 in the following way: Give this port line to the input of an 74HC32 OR-gate. Use a pull-up of 10k here. Connect the output of OR-gate to the /OE pin of 74HC595. Use a 10k pull-up here.

Take a reset controller with brown-out detection (e.g. MAX1232), connect the output of this reset controller not only to the reset pin of micro, but connect it also to the other input of OR-gate. Use a 4k7 pull-up here.

During power-up and power-down the reset controller is emitting logic high. Also, after power-on reset PORTx,y is emitting logic high. In all these situations the 74HC595 outputs are in tristate and the 2k2 pull-downs force low level at the inputs of ULN.

Toggle PORTx,y low only after the 74HC595 has been updated and contain correct data!