Making students hand write their essays

I teach 12th grade humanities, and in my classes, students do a fair amount of writing. Our school does not issue Chromebooks; students bring their own laptops to class, which as you can guess raises all kinds of frustrating issues: texting, scrolling the internet, online gambling, TikTok galore, ChatGPT usage, etc. Every time I've talked to my admin about this, and stressed how much of a problem it is, I'm told it's my job as the teacher to monitor their computer usage as best as I can. (Lol, yeah, OK.)

I've decided that moving forward I'm going to require students to handwrite as much of their work as possible. The problem: so much of their handwriting is atrocious. Some of this is due to dysgraphia, which allows for accommodations, but for many of them, they just don't want to pick up a pencil and write. They rush through, don't take it seriously, don't think it's "real" work if it's not on a screen.

Nevertheless, I would like to have their semester exams be entirely handwritten. I'd also like to spend some time in the weeks before having them practice writing out longer answers on paper. I want to even start grading them on their penmanship - mostly to get them to slow down, take their time, not rush, and practice organizing their thoughts on paper. (Again, in cases of dysgraphia, I will of course provide accommodations - though the sad reality is that the students who will need these are some of the worst offenders when it comes to ChatGPT and distractions. Sigh...)

In my estimation, going back to hand-written work is the only way to ensure they aren't being distracted by dopamine hits or using ChatGPT to write their answers (which is an insane problem at my school that no one at the top seems interested in addressing.) I literally cannot spy on the computer screens of 15+ students at once. And until we get our own Chromebooks, where we can block certain websites, or simply turn off internet capacity, these behaviors will never change.

What do y'all think: is this a hill worth dying on? Or am I setting myself up for impossible frustration? Am I better off just washing my hands of this issue and accepting the fact that I cannot control what they do on their own computers? That until my admin wakes up and realizes the problem they're creating, students will just continue to do what they do?

Bigger picture, I'm at the point this year where I have to decide what's worth swimming upstream for, and what's not. At the end of the day, my conscience feels rotten allowing students to get away with things I know will hurt them in the future, but I also have to acknowledge when it's just not within my control.

Thanks, everyone, for any insights.

Quick edit: sorry, I should've given more context about my school. Without getting too much into it, I teach at a small independent school that doesn't give consequences for student behavior (I'm not exaggerating; I could write an entire dissertation on this.) So even if I caught them doing something they shouldn't, the best I can do is give them a "redirect," document if they don't comply, take off participation points from their grade (which they don't really care about), and hope the admin follows through (which they won't.) I'm also not allowed to confiscate their laptops; parents would absolutely freak. I know that sounds cynical, but it's the reality of the situation. 😕