Hi nibler,
An example............ I have final orders in place from 2015 giving 7/7 shared custody of my sons with mum and equal parental responsibility (standard stuff). A few months after the Judge's stamp dried, mum text me one day saying she had to leave town 'for a bit' and could I take care of them 100% until further notice. After a few weeks of texts asking her when she would be back which she didn't reply to I contacted CSA as I was paying her at that point even for 7/7 to update the change in circumstances. It all created a paper trail as others have pointed out above as both CSA and Centrelink obviously had to contact mum for verification which she did eventually after a few months.
Be positive firstly because you can now give stability to your child which it seems they desperately need. Don't sweat the details feeling you need to prove she relinquished custody because if that time comes that she contests it then there will be enough evidence to support it via the school, family members, neighbours, sports clubs etc etc.
All of that will be basically irrelevant if your child starts to do well in a stable environment because that will be what the Judge would be more concerned about if she changes her mind in the future (don't forget she would have to take you to court to have the orders enforced again as you have full time care now)
Don't over think the game because you miss the end goal and that is your child being happy and healthy and mum just might realise that the best chance for your child to have that is with you.
Cheers