https://www.dropbox.com/developers/start/setup#android
I think documentation is very clear about what to do integrate dropbox in your application.
Does your manifest file contains
<activity
android:name="com.dropbox.client2.android.AuthActivity"
android:launchMode="singleTask"
android:configChanges="orientation|keyboard">
<intent-filter>
<!-- Change this to be db- followed by your app key -->
<data android:scheme="db-INSERT-APP-KEY-HERE" />
<action android:name="android.intent.action.VIEW" />
<category android:name="android.intent.category.BROWSABLE"/>
<category android:name="android.intent.category.DEFAULT" />
</intent-filter>
</activity>
and also internet permission?
<uses-permission android:name="android.permission.INTERNET"></uses-permission>
dropboxunlinkedexception occurs when you try to download or upload a file without successful login. Check authentication part here https://www.dropbox.com/developers/start/authentication#android and this should sort your problem