The MB nomenclature for this part is the guide rod. Most people call it a strut rod. Yes, if loose, this would certainly cause the problem you describe. I've had so little trouble with the front suspension of a 123 car, I don't know if this is a very common failure. The best thing is to get the car up where you can shake and pry around on the components to determine where your problem is. Don't do this unless you have experience with raising a car to safely perform these tests. If I encourage you too much to jump under the car to do this, I will receive the "wrath of Deezel", (the mercedesshop safety officer) and rightfully so. We have no way of knowing what equipment you have to work with, or what your experience level is, so we don't want to encourage you to do something that may prove unsafe.
But the only way you can know what to replace is to shake and pry. Be careful.
'84 Euro 240D, 516K miles
'88 300E 5 Speed
'81 300D Daughter's Car
Over 800,000 miles in