Typically there is NO perceptible axial play on the output shafts. You have a lot of play.
Have a close look at the image you posted with the arrow to the diff. The arrow points to the diff side and spider gears. Immediately to the left of the left side gear is the shim which controls the output shaft's axial play. Here's another image:
http://www.rdent.com/manuals/europa/tcparts/transmission/FM.gifThe diff housing (incorporating the ring gear) is a solid assembly. The external nuts set the ring gear's position and bearing preload but would have NO effect on the output shaft axial play.
The shimming people are talking about is the placement of shims between the pinned half-shaft yoke and the collar which the seal runs against. The seal collar presses against the very large diff housing and its bearings which have no problem carrying the suspension's loads. Without these shims, the suspension load goes to the internal side and sider gears. These are not up to the task and quickly wear giving the excess axial play you are experiencing. The play develops first, then complete failure.
Myself, I would remove the transaxle and repair the problem. This may involve just replacing the diff side gear shim, or fitting another unworn diff housing.
Mr E. TC is suggesting that you properly shim the half-shafts. This will correctly apply the suspension loads to the diff housing from the internal gears. The internal shim/housing/gears will not get "better" but they will stop getting worse. This may be worth pursuing and provide a lasting fix. Unfortunately it is impossible to know for sure without stripping the transaxle and checking the diff's condition.
So, now it is your decision which way to go.