I think failing transverse arms will cause slight directional changes to the rear wheels and so effectively some unwanted rear-wheel steering, not to mention tyre wear. Its adjusting these arms that adjusts all of the rear wheel alignment.
I think you woudl hear load clunks from the rear if they were that bad. They would have to be very knackered indeed to cause wobbling to the extent that you describe but do I detect that you are saying things have improved a bit?
You have two front/rear transverse arms and two rear/rear transverse arms. As usual its not your parts its your labour, mate. When I changed some of mine on my 156 the only way to get two of them off was to drop the rear suspension bar (subframe?) as the long fitting bolts foul on the petrol tank. I don`t know if the 147 suffers from the same, it depends where the petrol tank is. Either way, you must have an alignment check again afterwads as things will all be to **** after new arms have been fitted. If you are going that far then you may as well do the shocks too as you are half way there.
Don`t do this job yourself unless you are confident and good at these things, there are a lot of pitfalls with bolts that haven`t moved for a long time and if something goes wrong you have an undrivable car.
BTW <circles like vulture> I have one of those arms for sale. Brand new genuine alfa, yours for a fiver plus post
I`d have to check the part number to see if it was front or back. Things like this often turn up on Ebay for about half the price if you are patient. They are the same for the 156 and 147.
<Edit> Just saw your question about who could do this. Yes, any garage could do this, its not alfa specific work, but don`t forget the 4-wheel alignment afterwards. Its one of those jobs that takes 1 hour per side if all of the bolts come off easily and bl**dy ages if they are frozen up. You can help yourself by spraying WD40 or Plusgas all over both ends of the arms for days and days before it goes into the garage. Try to get the job done any time but a saturday as nearly all of the bolts used in this area are alfa specific and can only be replaced from alfa if the worst happens and one breaks.