The order of the object + possessive pronoun is not as strict as English (for example). Is "rodzinę naszą" acceptible? Does this constriction alter the emphasis or is it awkward sounding altogether?
it's acceptable but awkward. I think your best bet is to use possessive pronoun before noun.
Dlaczego "(My) Rysujemy swoją rodzicę" nie działa? Więc "swój" nie działa na zaimek "my"?
I'm afraid that "rodzicę" is a word that you've invented :) 'swoją' is ok, although it turns out that it was forgotten here. Added now.