Lots of sympathy Daisy. This is horrible to live through, especially as when you don't know what the underlying problem is it's so hard to remedy. Good it's not MS though! I suffered something that sounds similar but milder some years ago. Basically I spent most of my waking hours wanting to curl up in a ball somewhere and sleep, which, for someone who'd always been fit and active and lived happily on 5-6 hrs sleep per night, was a huge shock. I realised I'd never really known what it was to be truly tired before then. The first couple of episodes lasted 3-4 months but then they got shorter in duration and happily for me haven't re-occured for about three years now. Unfortunately I'm not sure I can offer much specific help. I never found out what the underlying cause was (I was living in semi-rural Africa at the time so couldn't get much medical help, advice or tests beyond basic blood tests that turned up nothing but at least ruled out any tropical nasties!) but I mostly kept running albeit mostly gentle and shorter runs. I didn't find running or other physical exertion made me feel any worse - on the whole I actually felt at my best when running and it didn't seem to have negative after-effects in terms of making me even more tired - i.e. when I tried not running for a week or two I didn't feel any better physically and less cheerful generally. I did sleep quite a lot more at night though and made that a priority.
Sorry I can't offer a solution, just wanted to sympathise, offer a glimmer of hope that it will hopefully all pass eventually (hopefully sooner rather than later, particularly if the medics can find a cause) and say keep running (gently) if that makes you feel better in general.