Help advice on UC/LWCRA/ADP and moving in with a partner

Hi there, I was wondering if anyone could please help me with information. I am currently on a single-person UC getting £357 a month after deductions living with my mum. I also receive the Scottish Adult disability payment of £434.20 a month. I have just been awarded LCWRA and I have just been offered a part time job of 10hrs a week on minimum wage which I really want to accept.

My partner wants me to move back to the family home to be with our 2 children which I want to do, although no matter how I search online I cannot find answers to our situation and what effect it would have regarding rent and council tax. My partner earns £1117.10 a month in her job. She receives £1319 UC a month after deductions. Also the Scottish Child payment of £213.60 for 2 children. Child benefit of £170 a month. She also receives a council tax reduction from a Band C which is 20%. My partner and children live in a private rented accommodation which is £850 a month.

I understand there are a lot of figures here, but if anyone could advise me on how this would all work if I moved back in with my partner and children I would appreciate it. Many thanks in advance.