Although the ULN2803A can drive 500mA per pin, the total current the chip can drive is (I recall) 2.5amp. Driving 5 coils would hit the total current limit of the driver.
And when energising a coils, you often get an initial current surge, so for frequent switching of 5 coils you might have an overheat (or fried chip) problem. Maybe not immediately.
Maybe split the load over two driver chips.