I often feel this way when I am anxious too, and I have wondered about the same thing mammo raises, whether I am depressed. However, I don't necessarily feel depressed, I feel overwhelmed in those times, and my mind seems to want to just cocoon up around those anxious thoughts and stop thinking about the outside world. Sometimes, I feel that resting and getting more sleep is the only way I can find peace, too. So maybe it is just my body's way of trying to shut off the signal overload from the anxiety. I don't know the answer myself, but maybe it is something to which you can relate?
When you are feeling anxious, have you been remembering to eat and drink regularly still? When I am highly anxious, I am so preoccupied, I forget (usually for 6-12 hours, sometimes for whole days) to eat and drink, and then I not only feel sooo tired but also weak and I start acting weird (forgetting, dropping things, can't get words out of my mouth, irritable, etc.). Then I have to finally remember - gee I haven't eaten in X long - and force myself to eat something. Usually I LIKE to eat!! It's maddening.
It sounds like you have developed depression, which makes us feel this way. It robs us of so many things to include energy, our ambition to do things we once enjoyed, and makes us want to sleep a lot. Are you on any medication for your anxiety? Depression often goes hand-in-hand with anxiety, so know you're not alone. Let us know if you are on any medication(s), as this may be a clue as to why you're so tired as well.