That handle is directly connected into a pulling rod that actuates the door opening lever. I assume that if it has play in it, it might have stretched or slipped slightly in the hole it seats into in the chrome handle.
If this were my 190E, I would take the handle panel off and inspect the interior of the handle and the rod itself for problems. It may just need to be re-seated. Removal of the panel is easy - For no power seat controls -
Remove 1 screw (door latch retaining black plastic piece).
Remove 2 pieces of plastic - one at door latch on side held in by aforementioned screw and the other above the interior door handle.
Remove 1 bolt - holds on top of handle
Slip whole handle console forward and out and slip off the connecting rod.
(With power seats, there are 5 more pieces of plastic and an additional 3 screws.) Email me, if you need to.
Anyhow...take it apart and look at it.
John J. Meadows
'83 300D 3.0L 260k mi.
'85 190E 2.3L 99k mi.