When you practice your organ piece, try to choose a slow practice tempo. The simple rule is this: play at the tempo in which you can avoid making mistakes. This usually means practicing at a half speed or 50% slower than the concert tempo.
The next thing you have to do is to subdivide your piece into manageable fragments of about four measures and practice them repeatedly over and over again. Also remember not to play the entire four-part texture right from the beginning because in most cases it will be too difficult for you.
What you can do instead is to play separate parts of that fragment. When this becomes easy, master it with two-voice and three-voice combinations and only then attempt to play the entire four-part texture. Always remember to aim for at least three correct repetitions in a row of each particular step.
In order to achieve and maintain accuracy in your public performance your mind also has to be focused. What this means is that you have to think about your current measure you are playing. Don't think about episodes or fragments that you have already played - they are gone already. Don't think about the future episodes - they have not yet come. Just stay in the current measure or in the current moment.
This will help you to stay focused and pay attention to details on a deeper level. Also this will help you to reduce the stress and the level of anxiety. You will not fear your audience and you will not fear any difficult episodes or fragments that are coming up if you perform in such focused manner.
Apply my tips in your practice and public performance and this will help you to maintain accuracy whenever you play an organ recital, a church service or just for your friends or family.