Was the course you did specific? It sounds like Son Rise to me? If so, you may find it difficult to find paid work as Son Rise seems to operate more on volunteers rather than paid (I was briefly trained in Son Rise for my long term charge, but his main time in the room was with parents and volunteers). If it was ABA you're more likely to find work as there is a demand for that (though it's still limited), along with Growing Minds, which seems to be becoming more known about in my area at least. They're the three I know somethig about and have seen advertised down here, the work is generally part time so it would suit your hours need. It's definitely worth checking out any forums related to the course you did and advertising on them too. I know that the ABA forums usually have sections for tutors, though it's been a while since I went onto them. Check facebook as well, as there are groups for people looking for work in particular specialities.
It may be worth listing the treatments/diets your child was on as well, there will be people following the same thing.
A lot of the childcare sites are a bit crap in that respect. The prices keep increasing. I found my initial live-out jobs through Childcare.co.uk and paid the subscription, but the families who contacted me were subscribed themselves and the price was much lower then, in my last job searches my local agency has found me work, but they refused before as I didn't have the 2 years nannying experience they wanted.
You would most likely be employed, it depends on the circumstances but it's difficult to fit the self employed criteria, which is difficult as on those sites there seem to be a lot of people who want "Self employed nannies". I asked families about their intentions regarding that before interview and didn't interview with anybody who wanted somebody self employed. Self employment is less to do with how many jobs and more about the criteria. I'm both employed and self employed. My employed work is regular, set hours for one family. My self employed work is ad hoc, I am free to turn it down and whilst occasionally I'll have a regular day of the week, it's only temporary, like this summer I've worked for three families, my employed job three days a week as usual and two days for other families for 5 weeks. If that makes sense? (I'm rambling).
What part of the country are you in? Nanny wages differ according to area. It may be worth calling your local agencies, they may not be able to find you work (or they might be happy to take you on! Criteria is different everywhere!) But they'll be able to advise you on what is going on in your area at the very least. Browse job ads as well for a rough guide :)