There's a thread here in this forum that has addressed this issue at length. I tried to find it for you but I honestly gave up after page 5. I hope you know a little something about digital media rights otherwise my jargon probably won't mean much to you.
When you download an a file, you are granted the "rights" to it. It's the program's way of checking whether something is legitimate and whether you can use it. The problem is that, for whatever reason, the game packages a lot of these little bonus items in "item packs" instead of individually. So imagine you download a file that contains items A, B, and C. When you clicked download, you really only picked item A. Items B and C just came along for the ride. By clicking A specifically, you acquired the rights for that one item only. Every time you boot the game, it checks to see if you have the rights to those items. If you don't, it connects to the DA servers to verify it another way. If you try to play offline, you'll get that error because it won't be able to verify you have the rights to B and C, because you only really truly downloaded A for real.
What you need to do is make sure you acquire all the digital licenses for all the DLC you have, so that you can play offline. It's relatively simple, just tedious. You gotta make a list of all the DLC items you have. You can figure that out from the DA main menu. Then you go to your Xbox dashboard, to your hard drive memory, to the dragon age folder, and write down which items are contained in each item pack. Finally, you need to individually download each item (from inside the game on that download content menu), then delete the item pack from your hard drive memory, then download the next item, delete the file, etc. etc. What this does is manually acquires the rights to each item in the pack so that when you play offline they'll all work.
I know this is a little confusing so I'll sum it up with a simple example. Let's say "Item Pack 1" consists of the embri's many pockets, the amulet of the war mage, and the blood dragon armor. (Just for the sake of argument.) When you go to the download content menu in DA and click download for embri's, magically the other two items will appear as well. But by clicking on embri's you ONLY downloaded the rights to use that one item. As long as you are on the internet you're fine. Try to play offline and it won't work. So now, go to your xbox's memory, find the dragon age folder, and delete the file marked "Item Pack 1". Then go back in game to the download menu and download the amulet of the war mage this time. Again, magically all three items will appear together without having to get them individually since they are in the same item pack but now you've only acquired the rights to embri's and the wage mage. Making sense? Finally, delete the file one more time and download the item pack again picking the blood dragon armor for the final time. By manually downloading each item in turn, you'll now have acquired the rights to use them all. The next time you play offline it should work fine.