Council tax statement? Anything from the council to show you've registered? It wasn't a requirement when I applied, so someone else confirm if this is required?
You don't show us the actual dates, so do the dates on your payslips cover six FULL months? From November to May should be fine but if, for example your payslips start November 30, 2020, and go to May 1, 2021, that isn't six FULL months, but five months and a day. But if it's November 30, 2020, to May 31, 2021, it's fine.
Do your bank statements cover the same FULL six month period as the payslips and show all your payslips going in?
I don't think they'll notice the address on the marriage certificate, and if they do they'll see the information about you completing the house purchase and moving. But since you're already including a letter to explain the lack of official correspondence/cohabitation evidence, you can just add a little in there.