@Darwin4Ever Welcome to the forum!
Bitwarden certainly is a decent replacement. 
But… it literally is an online-password manager. And besides the portable app that you already mentioned, the Bitwarden clients/apps indeed require a server to connect with (either with the BW cloud or self-hosted).
There is a feature request for “offline editing”, when the clients can’t connected to the server: Offline Editing (management of writeable vault items) – and there is a feature request for what you are requesting: Full local offline password manager (no self hosting or cloud hosting)
I think you mainly have two choices now:
- looking for another password manager that can be used only locally as you want it
- considering, if an online password manager – like Bitwarden – where the data that “goes out of your local environment” is encrypted all the time, might not be so bad after all