I switched from Keepass to Enpass a while ago (enpass.io). It does use 3rd party cloud accounts for synchronization, but it's fully integrated into the apps.

Has android/ios/blackberry/windows mobile clients, desktop clients for mac/win/linux/chromebook (including portable versions), and browser addons. It's not a subscription service---the desktop versions are free, and the mobile versions cost a 1-time purchase to unlock all the features. I'm very happy with it.