The Goblin War is detailed in novels rather than game supplements. Books 2 and 3 of the Cormyr Saga are about the situation leading up to, and during, the Goblin War itself, respectively. So for information on the events during the Goblin War itself, Death of the Dragon by Ed Greenwood and Troy Denning is the book to read. For information on the events directly preceding the Goblin War, Beyond the High Road (Troy Denning) is the book to read. And for all the history that precipitated the Goblin War and motivated the character who created and lead her army against Cormyr, start with book 1 in the trilogy, Cormyr: A Novel (Ed Greenwood and Jeff Grubb).
For material useful to setting an adventure just before the Goblin War starts, you'll probably want to read book 2 for information on the current state of history when your adventure begins, and maybe read book 3 to mine for events and characters that might be in the distant background, or even interact with the PCs.
You're correct that “Into the Dragon's Lair” doesn't contain any information on the War itself. In its introduction to the adventure, it says the historical background isn't necessary for DMs and directs the curious reader to books 2 and 3 of the Cormyr Saga for background information and a richer portrait of some of the locations visited later in the adventure. It might still be worth tracking down though, as it might be an interesting follow-up to your own adventure.
(NB I haven't read these books, so I can't comment on their actual quality as novels. The reviews at the Goodreads links above indicate that they are, at least, towards the upper end of TSR's novels' quality.)