Depending on where you’re going, you may not need to worry about it much. When I was in postsecondary education, there wasn’t much handwriting required. And I graduated 13 years ago; certainly things have gone more online since then. You might want to check with a current student in your field of study at your university and see what the handwriting requirements are. Make sure to ask whether cursive is a dealbreaker.

If it is something you’re going to need to work on, there’s really no getting around it: you’re going to need to practice. Get a big notebook, and something to write (hopefully something you’re actually interested in), and just start writing. Transcribe a TV show as you’re watching it. Copy a book line-for-line. You get good at the things you do a lot, and so you’re going to have to write a lot.

Also, I would recommend slowing down. My handwriting is great when I’m writing slowly but can be terrible when I speed up if I don’t pay attention. Slow down to start; if it’s still not legible, slow down even more. Make sure you aren’t practicing your existing bad habits. Then, as you practice, be deliberate: focus on each individual letterform, and as you become more comfortable writing legible letters, try to pick up the pace.

There are other things that you might find help you out: try practicing on wide-ruled paper, rather than college-ruled, for instance. Try a pencil or pen which moves more roughly across the page, for more tactile response. Make sure your pen or pencil is making strong, clear marks so that it’s obvious what legibility issues are your hand (and not just a bad implement).

You can change your writing style; I have, on a couple of occasions. It just takes practice.