One of the main differences between freehubs and freewheels.

• Freewheels, the axle bearings lie behind the freewheel.
• Freehubs (Shimano style), the axle bearings are inside of the freehub.
  The advantage of this is that it moves the bearings outward to give better/stronger support to the axles, especially with more sprockets and wider cassettes.

It is hard to tell how loose everything is from your description, or how long it has been that way. One option would be to simply tighten the non-drive-side cones and leave the drive side alone.

However, if it has been loose for any significant amount of time, it puts everything at risk for wear and breakage.