Just make sure to know exactly what the rules are. Many stores charge a fee for layaway. That would make it instantly not worth it to shop early. Instead it would be better to stash away what you would have spent that week.
Also, the closer to Christmas the better the deals.
And I don't know about your children, but at this time of year, mine often did not know what they wanted for Christmas.
If you are buying for other adults consider making food gifts instead of purchasing gifts. You save time and gas running around to the store. You don't have the headache of wondering if the clothes item will fit or if they will like it. And you will save a ton of money. Men, especially like food items. You could make homemade mustard, fudge, bar b q sauce, cookies, jar gifts where they assemble it themselves. You could even make little cookbooks to go along with the food.
I know for many folks half the fun of Christmas is shopping. I also know that shopping gives me the "I wants" and "I need," and "Gosh darn isn't that the cutest thing I have to have," and "It's on sale I can't pass it up." Just by not shopping I save a ton of money.