First of all, we must understand that singing a part implies that you don't play it on the organ. In other words, if the piece has 4 parts, you can play any combination of 3 parts and sing the remaining one.
When you are doing that, a curious phenomenon is taking place - you are developing your musical hearing, more specifically, harmonic hearing, which is one of the most valuable assets any musician can have. This means that you are acquiring your skills to listen for specific harmonies, because the organ texture constantly produces different chords. These chords could be openly visible as in 4 part writing or they could be hidden and implied in a solo part and 2 part texture.
If you haven't trained yourself in singing a line while playing others in a polyphonic piece, when you first do it, you will soon discover how incredibly difficult this task is. The challenge increases with the number of parts and with the rhythmical independence of voices.
To make this task easier, you don't have to start playing other parts right away. Learn to sing separate voices first. Just like learning to play a piece you work in small fragments on solo parts, combinations of 2 and 3 parts before putting all parts together, assume the same systematic approach while singing each line.
Do this experiment with any 4 part polyphonic piece that you are working on. Practice all voice combinations of 1, 2, 3, and 4 parts and sing any part in each of them. The key is working in small fragments of about 4 measures slowly and repeatedly - until you can do each step fluently at least 3 times in a row.
At the same time when you are doing this experiment, take another similar piece and learn it in your usual way without singing. I think the results will pleasantly surprise you.