RG172 shimmy
King RB - I had EXACTLY the same experience as you on the 172RG (in South Africa) and the shimmy was constantly being fixed and then recurring. It got to the point when I would include the shimmy in my pre-take off pax briefing(!) Always on T/O and cured with some back pressure, as you say.
(as an aside, I never figured out that spring loaded steering - on one occasion I returned to the ramp as I was sure the thing was not steering properly. Engineering them proceeded to drive it round in tight circles but were also unsure if it was OK. In the end I just flew the thing...)
The shimmy presumably has its origins in the spring loading on the steering.