Probably the most important tip here is to master each individual part separately. Practice it to the degree that you could play it from memory from the beginning until the end without stopping.
Apply the usual memorization technique where you subdivide the piece in fragments of 4 measures and first memorize each measure individually, then two measures at a time, then three measures without stopping and at last the entire four-measure fragment. In order to connect the fragments and the measures, start and finish each fragment on the downbeat of the measure.
After you master each part separately, start combining the two parts together. If you want for this process to be as enjoyable as possible, every single combination of two parts must be mastered before going on to the entire three-part texture.
Every single step in the above procedure has to be done very slowly. Perhaps twice as slow than the concert tempo. You will pick up the tempo very naturally once the piece has sunk in into your hands and feet.
Some people do these steps but still have problems with controlling the trio texture. If this is the case with you, there might be two reasons for this.
The first reason might be that the piece is too difficult for you at the moment. It doesn't necessarily mean you will not master it. However, it might take many months in which case you might lose interest in practicing them over such a long period. Therefore, it would be wiser to master a few less challenging pieces before going to the advanced level composition.
The second reason might have something to do with focusing your mind. Very often we sort of know the piece and it appears that we have done all the necessary steps to master it and yet we still make some mistakes. Sometimes we can play fluently, sometimes not.
Then try to fix your eyes (and your mind) on the measure you are currently playing. Dismiss your thoughts about the previous fragments or the ones that are coming up next. This is very difficult to achieve because our mind is constantly moving. But like every muscle, the mind can also be trained with diligent and mindful practice over time.
Please note that even in less advanced music written in a trio texture, many organists feel quite a challenge to control the three parts. So if you are in such situation, the above points and tips will definitely help you as well.
P.S. There could be a third reason as well but we would like to dismiss it sometimes overemphasizing the importance of simply playing the music. I'll say it anyway: the lack of practical knowledge in music theory.
If you could build a solid foundation from the ground up of how this piece is put together, if you could know exactly why this note (or chord) is here and what is its function and meaning in a musical composition, then not only your understanding and appreciation of the piece would improve dramatically but also your playing level as well.
I have found out that the best way to go about learning music theory and the structure of the composition is to created a similar piece based on the model you are currently playing. Not only it is absolutely amazing feeling to have created something of your own but there is no way you can do it successfully unless you understand all the intricacies of your model piece.
It doesn't mean you have to compose long and difficult pieces. It could be simple 8 measure preludes at first.
You don't need to look very far for the proof of the above statement. Look at any master organist you respect. They know their pieces inside out. Moreover, they think like composers do. In fact, many of them are composers themselves.
So start today.