In other words, do you try to think like a composer who created this piece long time ago?
I think some people who would like to do this, simply forget about it, simply get carried away by the music. Why is this happening?
One of the reasons of course is our ability or inability to notice musical text and the details. When we first start playing the organ and when our music theory experience is low, we notice very few details about that piece. Perhaps recapitulation of the theme or something else that is very obvious.
But mostly beginners think about the notes.
Later, we can start to pick up some more intricate details, like keys, thematic entrances, sequences, cadences etc.
Very advanced organists can of course know all the musical material of the piece, they can see how the piece is put together right away.
I suggest you try to at least discover major tonal areas (tonal plan) of the piece and notice them while playing. Say to yourself, "this is Eb major or C minor or G major or E minor." Transposing a piece or a part of it to other keys helps a lot too.
Of course, you have to be really comfortable with the musical text, have almost memorized it. Only then you can start noticing things that the composer worked hard to hide from an average eye.