We might have a clue about this from a Fourth Age cycle of stories (that - so I presume, is one of our sole examples of the "new tongue" in the text rather than transliterated as English. There's little reason to believe a fourth Age text has an Old Tongue title, anyway...) It looks and sounds, of course, very much like the simplified version of the OT the new tongue is supposed to be:

The "Cycle of the Dragon" reads Charal Drianaan te Calamon.

It appears that over the course of millenia, the title of Dragon has evolved from the OT compound Calaman, that is: "red dragon". It's either that, or the OT name for Dragon was Calaman, and it's abbreviated into 'aman in a compound name like siswai'aman.

So, the OT title of LTT was "Aman" or "Calaman", and in the NT the word is "Calamon". The word "te" in the NT has replaced "an".

It becomes more obvious why the Forsaken aren't too mixed up and why the characters can't understand the OT at the same time. The OT used compound words with complex rules of word and sentence formation. The NT has set in stone the old compounds to form a fixed vocabulary, simplified the grammar a lot, and those new words have evolved over time so that many don't follow the compound rules anymore. The Forsaken know the rules of compounding, so it's much easier for them to puzzle out the meaning of the "bastard" words in the NT than for NT speakers to puzzle out how the OT compounds relate to their own vocabulary, especially that they have lost knowledge of many of the "root words", which forcibly (as in all compounded languages) often have a large palette of purely context-based meanings (an example of this is how the root word for "left" and "west" is the same one).