I only watched the original anime and then started reading the books.
I'd give the first three a try. Reading the first makes you appreciate how well the anime adaption did at bringing the story to life.
But be warned, the second novel is a bit dull compared to the first one, which is by far the best since it covers the most interesting part of Annes life.
And yeah, you can read it as an adult. Why not, if they are entertaining you.
Another warning, if you remember the end of the anime and the first book, there will be more of that in the other books aswell.