somehow newly saved contacts get disappeared
I have the same issue with my Moto G (3rd gen) with just one google account.

When new contact adding it will disappear. I resolved my problem by the following solution. The solution was to disable google contact sync. I fixed my problem. But I not sure what is exact cause for this issue. This is a workaround solution.

I hope this is a help if someone doesn't want to sync contact with google.