Welcome to the forum!

HIstorically AC is usually very heavily booked due to the first of the Alaska cruises finishing up in mid to late May. May & Sept cruises are popular because of their cheapness compared to the peak season in Jun Jul Aug.
Easter & School Spring vacation breaks are mid to late April this year so that shouldn't affect southbound loads in May.
I'd probably stick to LAX as you have QF NZ UA and FJ between them providing upwards of 6 flights per night heading south.