I THINK (emphasis on think) that you may need to make a workaround using the "move towards" function instead of the drag and drop behaviour. What I mean is, rather than the drag and drop behaviour, implement events that essentially execute logic similar to this:
While mouse is down
move object towards mouse.x,mouse.y,
That way clicking (and holding) can simulate drag and drop capabilities (especially if you adjust speed and such so that it does so quickly), but it still should interact with solid objects.
I hope this helps some. Good luck with your project!