1Password can sync your passwords through WiFi [1] without going through some cloud service like Dropbox. This is the main reason I use 1Password, and it so far has met my needs. Caveat is apparently it can only sync with a single computer.
Unfortunately I can't use 1Password at work because it doesn't have a browser client, and I can't install anything I like on my computer (corporate). Sticking to Dashlane for now.
Technically, a 1Password vault contains a "1Password.html" file that essentially loads up a browser interface to your vault. This may or may not meet your needs, depending mostly on whether or not you're cool with carrying around a thumb drive with your password vault on it.
[1] https://support.1password.com/guides/mac/sync-wi-fi.html