I would suggest setting it up so that auth is handled in the user browser, redirecting back to your own site. From there you can pass the oauth token to your app through custom url pattern yourself.
Another way would be to load the oauth page in a webview, catch it on your site, and inside your app read the webview url to get the oauth token.
You should never require the user to store their password in your app. An oauth token is the way to go, and is all you need.