I really don't think Latin is going to help with poor grammar and punctuation, given that latin grammar is very different to english, and there's very little punctuation. Take it from someone with 5 years of Latin (A-Level and then during Degree), it really doesn't have any effect on your ability to write in English. All it really teaches you is some interesting connections between latin root words and modern day english. I would suggest not adding to your workload with a new subject.
Is the problem that you can't spot your own grammar/spelling mistakes? Are you able to see them in sources you haven't written yourself?
You say you read a lot, and that would be my main suggestion, but reading purposefully. Just because something is a published work doesn't mean it has perfect grammar, so often you need to be reading and asking yourself 'is this correct grammar?', or purposefully reading things with bad grammar/spelling and rewriting them
Linking to that, my other suggestion would be writing. Write lots, even things unrelated to your A-Levels, and then purposefully revisit them to try and find mistakes, rewrite them until you can write them well the first time.
When you're reviewing things you've written, use something like Grammarly (free online) to begin with, but don't install the extension, instead upload the document with your writing in after you've finished - you want to notice the changes it's asking you to make, and decide if they're correct, not have Grammarly just do it for you so you never think about correct grammar.
Take auto-correct off your phone if you can, it'll make you pay more attention to your spelling/grammar in your everyday communications.
My big tip for spelling/grammar in exams is to stop after every paragraph and re-read it, rather than waiting until you've finished the whole essay to re-read. This means if you need to make big changes you've got the space to do so. On that note, if you're not already saving 5-15 minutes at the end of your exams to read through, you really need to if this is such a big problem for you.
Hope all these tips help