"I wonder if the quart bag has to be zip closed"
Yep, it has to close or they'll reject it. If they bother to check it in the first place. TSA is, inconsistent at best.
Two notes: first, the container size is technically 3.4 oz, but 3.4-1-1 wasn't as catchy, so they made it 3-1-1.
Secondly, one bag for each traveler. So you can each carry your own shampoo, toothpaste, etc.
I didn't have any trouble, and the secret is the same as you use to do only carry-on: you don't need as much as you think you need. :-)
Shampoo & conditioner: do you really need both? if so, get a shampoo/conditioner combo? Or buy the solid bar kind from a place like Lush.
Toothpaste: Baking soda? OK, I'm kidding. But unless you'll be traveling for three months or have an extra set of teeth, one small travel size of toothpaste ought to be plenty, even for two people.
Body wash: Yeah, I prefer it, but for traveling, soap was fine. Italy, especially, has some nice soaps too, so consider stopping into a store and buying a bar for use.
For that matter, you can buy just about anything you'd need, and shopping in the local pharmacy/stores can be a lot of fun. You can also take advantage of the exceptions, like the one for contact lens solution.