Although I understand the datasheet says a total of 70ma of current for the entire shift register, people everywhere seem to be using it just fine with about double that current (8 leds), so I'm wondering if it's worth it to switch to a more expensive version or if I can get away with going over the current limit. Any advice?
What you 've seen, is probably, led matrix 8x8, where HC595 driving 1 led at a time (20 mA). ULN2803 could buffers 595 to drive a column 20x8 = 160 mA , setting hardware cost for 3 IC overall. Nobody says 595 alone won't work setting 8 pins at 10 mA simultaneously, just abusing IC annulling any warranty for long term (~10 years) service.
It wouldn't be used much (or for 10 years for that matter), and assuming i half the current draw of the leds to 10ma a piece (given 8 leds), going over 10ma isn't as worrying as double