Well, there are various approaches and in every case the organist has to deal with a lesser or greater degree of fear.
A fear as in playing something which has never been played before. You don't know how it's going to start. You don't know how it's going to continue or end. Everything is risky. Every turn might end up in disaster.
And this fear can be decreased or amplified if you restrict yourself with a certain technique you might want to use. You will hardly feel any fear, if you trust yourself in that technique, as in playing a hymn tune in two parts note against note very slowly in your practice room and the other part ONLY uses intervals of the thirds and sixths in alternation. But if you say to yourself, "Well, I wonder how I would do with 5 part double fugue when the entire congregation is listening?", you know how the fight will end up, don't you?
One of the least amounts of fear involved is when you transpose. Think about it for a second. You have your hymn tune written out, as in hymnal and all you need to do is to play it minor or major second up or down. An approach which helps me a lot is to imagine or write out scale degrees of the new key on top of the soprano part of the hymn. Then you can just play those scale degrees in any key you want. Even in a different mode (minor, if the tune is in major, for example). That's all there is to it.
Try experimenting with transposition and see if that helps to get over the fear of playing what is not written on the page and lead into the real world of improvisation.
Diatonic Sequence in A Minor: IV-V-I