Here’s what changed my handwriting: turning it into a hobby
At school, writing stuff all the time was more like a job, so naturally I wasn’t really that interested in it. However when you don’t have to write a ton of stuff every day, you can approach it form a different angle. Go to the library and find a book about calligraphy. Pick a style you find interesting, and start practicing.
Motivation is key. Don’t write stuff you hate in a style you despise. That’s just basically a long way to say: school. Let’s say you fall in love with textura quadrata, so you start practicing that style instead of cursive. Eventually you’ll try italic, humanist, uncial and other styles too. Then you’ll start writing longer things; not just shopping lists, but journaling too. After a while, you start to notice that your handwriting has improved.
BTW don’t go with a dip pen at first. Instead, get a Pilot Parallel Pen, because it’s good for for most styles. Once you’re familiar with a specific writing style, you can start learning how to manage the ink flow of a dip pen, because that’s a completely different ball game.