Welcome and congrats on getting started. I commend you on looking for advice on how to help your family.
Once trust is broken it can take time to regain. I suspect they will continue to have some anxiety for some time. The first step to regaining trust is taking full accountability for what has been done. It sounds like you are already taking steps towards accountability. It also sounds like you have apologized for what you have done and you are in the process of overcoming this addiction. That is amazing and I am certain this gives them some comfort.
The anxiety they feel now is a normal human reaction. If they mention their concerns, validate them, apologize and tell them you are working to regain their trust. Only with time and hard work from you, can they eventually be less anxious. As their thoughts are justified and rational there is little they can do to change their anxious thoughts in this case.
They can do things that will help them sooth anxious feelings, journaling, mindfulness, exercise and relaxation techniques may help. You may also want to consider family therapy.
In summation, regaining trust will take time. Continue to show them you can be trusted and with time their anxiety will decrease. The most reassuring thing you can do right now is understandng why they are anxious and giving them space to work through those feelings. Healing takes time. It sounds like they have been patient with you, now it is time for you to be patient with them.
I wish you and your family all the best.