Yes, and even if a doctor says a symptom is due to ME/CFS, it's still worth maintaining some skepticism. As @Spartacus says, some doctors are happy to attribute just about everything to ME/CFS/hypochondria.It is best not to attribute any symptoms to ME/CFS unless advised to do so by a doctor.
Limbs falling asleep is not a recognised feature.
But for this particular symptom, I do suspect it is part of my ME/CFS. when I and my two children developed ME/CFS at the same time, we all started finding that we would wake up with 'dead' limbs, and get pins and needles when sitting and even when eating and driving (holding arms up at chest height). It was something we remarked on independently, before we knew that the others were experiencing it too, and it happened much more often than normal. It still happens. 20 minutes of driving can bring on pins and needles in my arms, I end up driving with alternating hands, one hand on the wheel while the other hand slightly recovers resting on my lap.
Yes, I too have measured very low pulse pressure. I've measured shock index too, and that is low when my symptoms are worst. I can get very cold hands that don't warm up. My son and I get Raynaud's syndrome. I get fingertips that I can press and the skin doesn't bounce back to refill the dent. My son's lower legs are very large/swollen, and his face goes pale when he has done too much. I am sure there is a circulatory issue going on, and it wasn't present before the onset of ME/CFS. I think this is a clue to the health condition that I and my son have.I also had low pulse pressure at the time. It was obvious something was wrong with my circulation. These symptoms suggest that there were local deficits in regulation of circulation.
Limbs falling asleep and associated issues may not be a recognised feature of ME/CFS, but I've seen them mentioned repeatedly as abnormally frequent by people with ME/CFS.