Well, it's been a long time since I've been so sleep deprived, but I say honor your body. General lack of energy is a reason to get out there and push, but exhaustion is your body telling you that you need rest, even if your head thinks you should be doing something else.
Maybe try a short gentle yoga practice after work, or before bedtime on work days, 20-30 minutes of centering and alignment. That will help work the worst kinks out of your body and might help you get the rest you need. Then you might have more energy for a gentle ride, hike or jog on non-work days.
As far as the night shift, I don't know what to tell you, because I know my body well enough to know I just couldn't do it. Not everyone can. Hope you can adjust. Light is enormously important to our circadian rhythm, so it might help you to make sure that your sleeping area is as dark as you can possibly make it when you sleep, black out your windows etc., but then light on waking is equally important, so you might invest in one of those progressive light alarm clocks, with natural spectrum bulbs that simulate sunrise. I've never tried one of those, but if I had to work night shift it would be the first thing I'd do.
ETA - might try taking melatonin before bedtime too, for a little while anyway. I keep reading that it really is not possible to "catch up" on sleep, so the more you can do to just get to sleep as soon as you can after work, and save the rest of the chores for non-work days, I think, the better. How long have you been on night shift?
Good luck and hugs.
ETA: just read this piece on ChiRunning.com and thought of you. It's pretty specific to running, but it talks about ways to get a great workout out of a run without making it terribly strenuous.
Last edited by OakLeaf; 05-15-2014 at 11:30 AM.
Speed comes from what you put behind you. - Judi Ketteler