How to set a new value in a paragraph field on submit?
I'd imagine the first didn't work because your custom submit handler is run after the form default submit handler so it's too late (You didn't provide this code so can't be certain). You'd need to make sure your submit handler is run before any other submit handlers. To do that in the array of submit handlers your needs to be first in the array

Similarly for the 2nd option you've tried, you're in hook_presave so you're changing things but then when the normal entity save function happens after this any changes you made are overwritten