Just to expand on the point ksand24 has made, the correspondence doesn't have to have both names on the same piece (though it's easier if it does). If you have correspondence in your separate names, you just submit one for you and one for your partner, and that counts as one of your six.
So, you have the jointly-addressed council tax statement. That's one.
Bank statements for each of you in June? That's another one.
A letter from your GP and an electric bill for your partner in January? That's number three. Etc.
Do you have stuff like that?
Edited to add: Okay, I admit I skimmed your initial post and missed the bit where you said you only just moved in together.
Ksand24, wouldn't they need to show how they maintained contact between November and June? Call logs, etc.?