We present a new H I modelling tool called GALACTUS. The program has been designed to perform automated fits of disc-galaxy models to observations. It includes a treatment for the self-absorption of gas. The software has been released into the public domain. We describe the design philosophy and inner workings of the program. After this, we model the face-on galaxy NGC 2403 using both self-absorption and optically thin models, showing that self-absorption occurs even in face-on galaxies. These results are then used to model an edge-on galaxy. It is shown that the maximum surface brightness plateaus seen in Paper I of this series are indeed signs of self-absorption. The apparent H I mass of an edge-on galaxy can be drastically lower compared with that same galaxy seen face-on. The Tully-Fisher relation is found to be relatively free from self-absorption issues.