Ring the Childcare Hotline and they will put you in touch with whichever organisation coordinates FDC in your area. There's usually a central coordinating office, they give you a few names, you go and interview and select the carer that's best for you. Easy!

BTW I have no problems with rejecting an unsuitable carer, I went against my instincts with a carer once because they told me there were no places (which later turned out to be hogwash, that coordinator left the job a short time later) and eventually terminated care for a number of reasons. If the person they give you doesn't seem 'right' for whatever reason, tell the office the reason and ask them to find someone else. Word of mouth is also good, if you know someone who's really happy with their carer you can always ask if they have a place or if they know another good carer who has a place. Keep trying and you will find a place that is right.