As someone who lived in Queenstown and Zeehan for 6 years of my childhood, I would advise avoiding the west coast. It's very isolated. My mum struggled a lot when we first moved there, we spent the first year driving 4 hours on awful roads each way on the weekend to get back to Hobart. While she ended up loving it, and we had a pretty good early childhood there, when it came to high school and education options we left again.
While there are not many places as cheap as there, there are areas that are still cheap but closer to better services. Wynyard, Ulverstone as mentioned by PPs are lovely.. If you need access to specialist services, however, as a nurse I would suggest being closer to Hobart. There are some lovely towns in the Huon.
It also depends on where you come from. If you are coming from a city interstate you may still struggle in some of these places! When we returned from living overseas (and interstate before that) we went to Hobart and found that difficult enough at first. But that was pre-children, and if you are particularly looking for the quiet life I guess you're ready for that
You will definitely get something for $500 per week, even if you decided not to go rurally! Good luck OP