Hi David,
Erica nailed it mentioning the importance of inks, nibs, drill practice and training yourself to keep your arm light and try to keep movement in the shoulder. It really is a combination of these things...along with the type of paper you use...but you already are using a great combo for obtaining fine lines: iron gall (I presume that's what the medieval ink is) and Rhodia.
Another important factor I would like to add to this list of factors is time. You have been only at this for 5 months. You are learning something that involves mastery of two key components: a new tool and a new hand/script. It takes a long time to build up the muscle strength and stamina to keep your arm light for long periods and to improve the 'brain to hand' directive. As you get more familiar and comfortable with the tool, hairlines become naturally thinner as you master applying and releasing pressure evenly...something I still struggle with 5 years into this.
Also, do not forget about the psychological component of learning. If those letterforms are not etched in your mind, your brain is going to be less certain on exactly how it should direct the hand/arm. You might have the 'gist' of the shape, but might not yet have a grasp on the finer details...and that uncertainty shows up in the form of shaky lines, odd shapes....and a heavy hand as physical reflex to mental uncertainty is to tense up, creating the 'death grip' on the pen...which pretty much guarantees heavy lines.
