Ok, this is MY situation, but this is how I did it.

I moved interstate. He tried to tell me before I left that I couldn't go til we'd been to court. Bull****. I found out my rights. He can't stop me leaving, but he can apply for a recovery order, which would mean I'd be forced to move back.

1, my ex didn't care enough to bother
2, when I spoke to the police here about that, he said the chances of the recovery order being approved were very unlikely, because ex knew we were leaving & let us go. A recovery order is only if you disappear as far as the sergeant here explained to me.
3, the longer they leave it to apply for a recovery order, the better your chances are of staying where you are, as your kids begin to make a life there. But you have to take into account what might happen if he were to take you to court. If you get the OK to stay, you might be up for full costs of travel for the kids to keep up a relationship with their father. Travel interstate isn't cheap.

Knowing whether or not to apply for full custody is the hard part. If you go to court, & there are orders put in place, you can't leave & have to stick to those orders. But you're securing your guardianship of the kids.
If you don't go to court, there are no orders in place, so you can go where ever you like, but as mentioned, he CAN walk in & take your children as he has equal rights. This is when YOU'D apply for a recovery order.

Its a lot to think about. Its impossible to know what's the right thing to do. I got lucky, my ex didn't particularly care, had no idea & was too lazy to bother finding out.
Not sure what it is in QLD, but DoCS/DCP can help with these situations. I used them when I left ex 5 years ago. I did go back but they still helped. They hired a trailer for me to move my things.