Rent Difficulty

So we started a separation. Still live in a same house but lead different lives. We've got one child. I started to look for rental property approx four weeks ago.It's been so frustrating all those weeks. Most of the ads on TradeMe say ideal tenants: professional couple. I apply and get a standard letter "your application unsuccessful". I'm an accountant, I can fully support myself and my child. Why is it that the landlord doesn't even consider a single mother with a child as a tenant? Our suburb is also difficult in terms of finding a rent, 2 bedroom rentals get thousands page views and the properties just go in one day. I went today to see another place. This was listed by owner, I emailed her yesterday and said that I'm mother with one child. She asked what time would be best for me to look, I said 12pm, she replied all great, see you tomorrow. I came there, and there is someone else to look at the property - a professional couple. So we went to look through, it was lovely, reasonable price, everything good. Then we all came downstairs, the guys asked when it will be available, etc, and then all of the sudden she said to them if you ready let's sign now. She didn't even ask me or them who we are, what our income, whether we have references. She smiled to me and said sorry. I was standing there like an idiot, in total shock. And then to add things up the moment she said sorry the guy started to laugh, like all over the place, literally, if you were on the other side of the street you could hear him. I went home, crying while driving, I just couldn't stop it..It's not that I look unprofessional or something, in fact quite opposite,it's just they don't seem to want a single woman with a child, or just simply don't want a young child in their properties?
Not sure now what else to do. I tried to put a notice at local library, but the stuff at library thought it's not what they can hang on their notice board. Although I saw all sorts of notices there starting from lost cat to single woman searching for a partner and yet the notice about rental seems to be inappropriate.
We've been living here for 10 years, our child was born here, went to coffee group, kindergarten, school, his best friends live here, I wouldn't want to pull him from school and his friends and move to other area (we are at St Heliers in Auckland) What else shall I do, shall I find some male friend who would act as my partner and pretend we are profeccional couple? Shall I become dishonest, and find some tricks? I'm just very upset after today experience, need a word of support from you ladies...