Welcome :) I think it's a great idea to get started on other aspects of your application before taking the LSAT...especially LORs because you want to make sure the professors remember who you are so the letters can be as solid as possible.
Although you don't need new letters, it wouldn't be a bad idea to ask your professors to update their letters. It's not a heavy lift for the recommenders—they'd only need to tweak a few sentences, or even a few words—and it would look a bit better.