I'm 52, my period stopped about ten months ago and I'm experiencing similar discharge, although mine only lasts a few days at a time (as it did when I was still ovulating). I've been told that it is actually possible to ovulate without having a period, and I did perform an ovulation test with my fertility monitor a few month ago, and it showed the normal progression of hormones until it indicated ovulation. I was told that a false-positive can be caused by post-menopause hormone changes, but in that case,the fertility monitor should have shown a consistent surge, not a gradual increase. I'm not sure if this is applicable to your situation - especially since your discharge has been continuous. I hope someone out there can provide us with information...
(P.S.: I've also had the ph-induced 'infections/irritation, but the discharge is typically not egg white-like in appearance/texture.