In order to be able to play organ accompaniments every week without stress and anxiety, you have to be able to sight-read really well so my first recommendation is to develop your organ sight-reading skills. The best way to achieve fluency in sight-reading is to practice playing new pieces regularly.
However, remember that you have to be systematic about sight-reading. Otherwise this practice will not work and you will not develop the skills that you need to play organ accompaniments easily.
Take a collection of organ music that you love and start playing one page a day. Don't play all the parts and voices right away because most likely that will be too difficult for you. What you have to do is to try to play just one single line and do this for the entire collection for a few weeks.
When this will become easy, go back to the beginning of this collection and start playing another line or part. So little by little you will be able to play the entire organ collection by playing just one line.
The next step would be to play two parts or voices at the time and later three voices and finally the entire four-part texture. Remember to proceed to the next step only when the previous step will become easy.
Do this in your every-day practice for 15 minutes a day and in short nine months you will easily achieve the level when you can play any organ accompaniment during your church service fluently and without stress.