I am a lecturer and I used to have one student with dysgraphia, it was me that suggested they had it as I noticed not only was their handwriting illegible, they also missed out words from their sentences. They also struggled to read back what they had written, say, a week later.
I would seek a private diagnosis with a specialist who knows about it; it's ridiculous that their online screening cannot possibly assess for that unless they saw their handwriting and assessed it properly.
You could, if inclined, be quite difficult about the fact that they are unaware in disability services about dysgraphia, that's poor on their behalf. It cannot be that difficult to screen for it, and the accommodation needed is so basic (a computer, like many students have) that it's not worth them quibbling about it.
You cannot just practice handwriting to solve dysgraphia any more than telling my severely dyslexic daughter to practice her spellings works. It's a learning difficulty, that's the whole point, and it won't be magically fixed by telling someone to just try a bit harder.
I would be seeking a private diagnosis as a matter of urgency and then reapproaching them with a view to firmly asking for reasonable adjustments.
There are many courses now where handwriting is never needed so you are a bit unlucky on that score, luckily computers have been invented!