Wow, this is confusing, and I'm a benefits adviser with a housing association so am used to this shit!
Because UC is paid in arrears, a rent account where everything is paid on time will still show 4 weeks rent arrears when a payment is due. The only time the rent would look like it was paid up to date is on the day the payment is credited to the rent account. And that's when everything is running perfectly.
There have been problems with UC rent payments being made to councils and HAs with missing or incorrect information, so that the payments can't be allocated to the right account. That can take ages to sort out. Then there's the problem with the SAP system referred to above, which has caused real headaches for some big HAs.
You need to go through your UC records and compare the payments the DWP have made with the ones that have been credited to your rent a/c, and see if any payments have been made but not appeared on your rent a/c.
I'd hazard a guess that you got a bit in arrears at the start of your tenancy because of thinking housing benefit were going to cover it, and that the rest of the arrears is just down to DWP paying in arrears.
When you moved, did you move to a different council area? If you're still under the same council as before you moved, you should never have been switched to UC anyway, it should have just been treated as a change and you should have stayed on housing benefit.
I assume that your move happened before the baby was born, and that you only had 2 children back then.
You could ask your HA have a financial inclusion team that helps tenants with this sort of stuff. They'll be able to sort it out much more quickly than you could!