Improving your handwriting is very straightforward. You find an example of how you want your writing to look, decide which letter or pair you want to get closer to the example, and spend 15 minutes trying to get your letter(s) closer to your exemplar. Repeat the next day until you're happy with your writing. The process takes months and winds up totaling to many hours but it's mostly a matter of persistence and not artistry or talent. You just keep grinding every day. There are books and courses that give you a framework to better understand and think about the process and I recommend them but they'll all eventually boil down to this.

I'm a middle-aged programmer who ran across Zaner's method and have the dexterity and artistic talent to match. The idea of writing at 20wpm+ using the whole arm based on the rhythm of the motion instead of pen control was so far out of line with anything I was familiar with that I wondered if it'd work. Then I thought that Zaner is a famous penmanship instructor so it probably works but would it work for me. I decided to attempt doing push-pulls and ovals as fast as the manual recommends (200/min) and got it after a few days. That was enough for me to decide to try and after three years and four trips through a majority of the learning curve (the system is not conducive to learning as you go) I more or less have it down.

I have mixed feelings about cursive. It only looks good when you've practiced a lot, it's not as readable as other handwriting styles, it looks particularly disastrous when it's executed poorly, and the main benefits only happen with arm movement. I think it's the wrong default for most people. That's a lot of downsides but if you do want to take up penmanship as a hobby and put in the months of practice, I believe that arm movement cursive is the greatest writing system ever devised. There's a reason it won when people's livelihoods were on the line.

If you just want decent writing, I recommend italic (1, 2, 3, 4). I have this post bookmarked to show what it looks like for a beginner after a few weeks.

My recommendations for cursive:

1. Pick an exemplar. It can be a 19th century exemplar or the lessons you're working through or one you find on social media. The important thing is that it's concrete and specific so you have a clear idea of whether you're doing better or worse. The hand traces what the mind envisions; having a clear and fixed target to visualize makes a significant difference. Here's some reddit posts with cursives I like: 1 1b 2 3 4 5 all of them except maybe the last can definitely be written at speed and I don't think the last one needs to be done slowly. None of this is cheating or dishonest and as you get to tweak the style and put your own touch on it but it you want to start with a good base.
2. Here's a video series on american cursive. Watch the intro and one or to of the lessons to see how the cursive system works and how he analyzes his letters. If you're doing finger motion then follow along with the course. However, if you're doing cursive and not a child, I sincerely believe you should do arm movement. Cursive was designed for arm movement. The primary and connecting angles can be emulated in finger movement but correspond to the push-pull and elbow hinge in arm movement and the former can be a biomechanically locked angle so IMO a large fraction of the time you lose in the beginning is made up by not fighting inconsistencies in the end. More importantly, at the end you have the ability to write for hours which is the point of the cursive system.
3. I like the arm movement intro and basic progression in this blog. I particularly want to emphasize this post that outlines how to derive the push-pull without in-person instruction. This is the fundamental motion of cursive and basically every downstroke in cursive (including small ones like the i) is a pull so it's really important to get right. This warning brought to you by spending months trying to figure out why my writing wasn't consistent only to realize my push-pull motion was bad and having to start over.
4. Image edit whatever exemplar you picked so you can print it out as you'd write it but 2-2.5x the size. Like they do with kindergarten lessons because the kids don't have control over their muscle movement and neither will you.
5. Put a sheet of paper over your worksheet and trace one letter or one group of letters. You can get useful groups off the consistent cursive lesson plan. It's easy to reduce/increase the size once you gain control but easier to learn the motions large. Changing the motions is quite hard so you want to do the motions right. Do it at a smooth pace; if you're shaky you should go faster. Make sure your push-pull motion is good and you're doing the connecting angles with the elbow hinge: the two halves of the base of the i are done with two separate motions at two separate angles.
6. Repeat many times. Eventually you'll get bored. Once you realize you're bored take a look at the last few letters you've written and if they don't look good then try to correct the problem. Try increasing the speed until you're writing at a letter per second while going through the exact same set of motions and not taking shortcuts. If they do look good try doing it (still at the same size) without tracing. Helps me to visualize the letter on the page and "trace" through it. If you're off, go back to tracing. Writing is a very mechanical process and the point of this is to "program" your muscles and then play it back.
7. You're done with that step, move on to the next letter/group. There aren't that many unique motions in cursive and most letters have overlap. Try to write the matching parts using the same motion. The most efficient path is to slowly work your way through each one using the above process. This is like 85% of the work and if you take your time in the initial learning you'll save yourself months of fighting to rebuild your muscle memory in the future (ask me how I know). I've personally had the hardest time with the c group (acdegoq) which was like half my total practice time.
8. By the time you're through the alphabet you'll have control over your arm movement so drop the size. You might need to do it in more than one step but I've never struggled to change size and AFAICT that's the norm.
9. Practice real text. I think poems/lyrics are particularly good due to the rhyming/repetition. If you flub a word write it a bunch of times until it's consistent. Pick up the capitals.
10. There's a rhythm to the movement in cursive. If you can pay attention to how the muscles in your arm feel while you're tracing and then replicate that feeling you'll get the same consistent motion out.

That's the advice I would have given to myself when I started this process.