What is the correct thing to do please advice.
1) The correct thing to do is whatever the two of you can agree on. In other words, there is no one universally "correct" thing to do.
2) If you have a common account into which everyone
the money they earn, what difference does it make whether you split the bills down the middle or your partner pays more because of his son? It seems to me that the whole point of a common account is to implement a "what's mine is thine" arrangement. If you want to continue tracking which partner pays for whose expenses, why the common account?
3) From what I think I'm reading between you're lines, you're not quite ready for the "what's mine is thine" approach, at least not yet. I'm not judging that one way or the other. But given that that's how you feel, I think it would work better for you if you and your partner kept your accounts separate for now.