The developer settings is just to get the phone detected by the adb so that you can debug your applications. Just because you did that doesn't mean that you can access all the folders in the phone. You will need to root your phone if you need to access the data/data folder in the DDMS. Or you could try to do this if you don't want to root the phone:
If the application is debuggable you can use the run-as
command in adb shell
adb shell
run-as com.your.packagename`
cp /data/data/com.your.pacakagename/
You can use Android's backup function.
adb backup -noapk com.your.packagename
You will now be prompted to 'unlock your device and confirm the backup operation'. It's best NOT to provide a password, otherwise it becomes more difficult to read the data. Just click on 'backup my data'. The resulting 'backup.ab' file on your computer contains all application data in android backup format. Basically it's a compressed tar file. This page explains how you can use OpenSSL's zlib command to uncompress it. You can use the adb restore backup.db command to restore the backup.