The joys of technology ruining our writing habits.
Do what Sandrie said. At the same time, don't use any electronic item when you take notes or want to jolt things down. Carry around a small notepad or notebook, depending on whether it's just an errand or class.
While doing that, pretend that you're writing this note for someone else. Kind of like when a TA takes notes for students who are absent. By doing that, you're disciplining yourself to practice writing each letter properly.
As for your signature, that requires you to know how to write cursive. No one's signature is perfect the first time. Mine was a mess 6~7 years ago. You'll eventually have to sit down and start scribbling your name in different ways to see which one you like the most and is the easiest for you to sign with.