You have described my problem exactly! Just over the past few weeks, my battery is draining very fast, and Google Play Services seems to be a cause. I've got very few apps installed on my phone, and I've checked them all but found nothing to indicate problems of a sudden.
After more searching, I found others reporting not only this issue but loss of 4G mobile, which then I noticed had also affected me. In reporting this to my carrier (CREDO, on Sprint network), it was suggested I try a profile reset. I had to repeat it twice to get everything working, but afterwards, I had 4G mobile back AND my batter usage went back to normal, which means going 2-3 days before needing a charge, whereas before the profile reset I was having to charge twice a day. Something you might try. To do the profile reset, I had to dial a sequence into my phone; I don't know if the sequence is the same for every network. Customer service for my carrier gave me the sequence to use.