[jmon]

That's different. Most shower heads usually unscrew and you just screw a new shower head on.

If it's all one piece as you say, then I don't know of any other way but to take off the entire shower arm and replace the whole thing with one that pivots.

As you stated, where it screws into the wall is very unstable, so being how you are renting this apartment and responsible for any damage, etc., the safest thing to do, imo, would be to get your landlord involved. Explain to him/her that the shower arm is very loose and water could be getting behind the wall causing damage. Maybe they will have it fix and you can get the shower head you want. Just a suggestion.

Others will be along with more advice/suggestions.