If you have programs installed on the D: drive you can't just move them, it's the same issue if you just copy the data to an external drive. The path is set in the registry and the programs will fail to run.
To just copy the files though, select them all, hit Ctrl-C, click on the E drive and hit Ctrl-V
Have you checked if you can free any space on the C drive by deleting TEMP files, maybe moving the swap file over? I don't like partitioning a single drive into multiple drives for this exact reason, almost every single time you end up needing more space than what you planned for.