Unfortunately, desire for cheap solutions doesn't change reality. I can tell because one out of five cars I have seen for thirty years comes with that caveat.
It sounds to me that you have a tierod problem, though. (which is the least expensive alternative). Lift the car and see if there is play when shaking the tire (holding it at 3 o'clock and 9 o'clock). There should be no play. Yours should be obvious.
I suspect the steering linkage rather than the box because of the changes when braking. The force of braking causes the play to go all the way in one direction which then changes the alignment. If the play were just in the box the alignment wouldn't change with braking.
Bosch Master, ASE Master, L1
33 years MB technician