I booked my trip back in August about 6 days before we left!! I didn't get my tags but I didn't expect to. I had only carry-on, so getting the bags wasn't a concern for me. I would make note of the claim numbers and put some distinctive ribbons or such on the luggage. If there is a digital camera handy, take a photo of the bags to show what they look like.
On my first trip using DME last June, I hadn't gotten my tags 4 days before we left so I made my own tags....cut yellow construction paper into 8"x3" strips. Put our resort name on it, DME on it, our last names, and '1 of 3' bags, '2 of 3 bags' and '3 of 3 bags' on it and the date of our arrival. Then I lamintated them. punched a hole in them and hung them on the bags. Wouldn't you know that my tags arrived that afternoon??!!!