If the Android project compiles correctly but is missing classes when run, it may be because ADK r17 (the current version atm) doesn't include the external game project as part of the build. There's an extra step you have to do that's still lacking from most of the existing documentation for project setup. It's described here. Copying from that site, the steps are:
Click on your Android Project
Select Java Build Path
Select Order and Export
Check all your references, e.g. the gdx.jar, the gdx-backend-android.jar, your main code project etc.
The pictures on the site are helpful, so go check that out.
Generally, this problem (missing GDX libraries at run-time on Android) is because the "libs" directory isn't setup exactly as specified in the project doc. The directory must be called "libs" and it must contain the GDX backend and both sets of ARM libraries.
The behavior is odd because since you don't get a compile error on Drop.java, it appears that you have your project set up properly. For the Dalvik DexClassLoader to fail, it appears that something is getting messed up in the build process. If you really want to investigate, take the .apk and run it through dex2jar and open it with a Java Decomplier and see if there are classes missing.
If the classes are there, it would suggest to me that you have problems with your development environment. Either Java/JVM, your Android SDK or the build of libgdx you have is corrupt. Since it works in the Java Project like you said, one would think that maybe trying to reinstall the Android SDK may resolve your issues.