i mean, honestly it’s amazing the Elves had as many languages and dialects as they did, considering Galadriel (for example) is over seven thousand years old.
english would probably have changed less since Chaucer’s time, if a lot of our cultural leaders from the thirteenth century were still alive and running things.
they’ve had like. seven generations since the sun happened, max.
frodo’s books are old to him, but outside any very old poetry copied down exactly, the dialect represented in them isn’t likely to be older than the Second Age, wherein Aragorn’s foster-father Elrond started out as a very young adult and grew into himself, and Legolas’ father was born.
so like, three to six thousand years old, maybe, which is probably a drop in the bucket of Elvish history judging by all the ethnic differentiation that had time to develop before Ungoliant came along, even if we can’t really tell because there weren’t years to count, before the Trees were destroyed.
plus a lot of Bilbo’s materials were probably directly from Elrond, whose library dates largely from the Third Age, probably, because he didn’t establish Imladris until after the Last Alliance. and Elrond isn’t the type to intentionally help Bilbo learn the wrong dialect and sound sillier than can be helped, even if everyone was humoring him more than a little.
so Frodo might sound hilariously formal for conversational use (though considering how most Elves use Westron he’s probably safe there) and kind of old-fashioned, but he’s not in any danger of being incomprehensible, because elves live on such a ridiculous timescale.