If you search on the web you'll find newspapers quoting a very specific day in advance. You can also find advice on this forum, since this is a regular question.
Do you know who some of the biggest purchasers of supercomputers are? Airlines. They constantly try to wring every dollar they can out of ticket sales. Weather, conventions, travel patterns, the economy, you name it and they try to use it to estimate demand. And it isn't as if the schedules are printed and sealed away for good. They are changing daily, perhaps even hourly, as they use current booking info to try to figure out how to sell the remaining tickets for as much as possible. That's why the magic day in advance changes in every newspaper article.
My advice is to look at the historical range of prices for that time of year, assume this year will be similar, and if you see a fare in that range or better then take it. You won't get the rock-bottom price, but you won't be buying an expensive last-minute ticket either. So, sorry, I don't have any tips for "BIG savings" but I imagine you'll get some replies promising when to buy to get that; of course if they're wrong, don't expect reimbursement.
Your choice to buy from the airline directly is smart. If there are any problems you want them to own the responsibility for fixing it, not to have them refer you to a travel website that in turn says "sorry, call the airline."