Problem solved - I hope!
Was chatting to a mechanic in the bike shop just outside the backpackers I’m currently staying at. He has had several recommendations on various chat forums and he did seem to know his stuff - and more importantly admitted not knowing a lot about my bike specifically.
Anyways, after a chat we narrowed the problem down to either the springs or tyres. I mentioned my thoughts that teh pressure in teh front was a bit low, so we measured it; with my gauge it showed 1.7 bar (about 25 PSI) - a bit low. With his gauge, it registered <20 PSI! The rear was the same story. This might also explain the wear pattern.
So we put a bit of air in both tyres and I took it for a run. Immediately - while turning and heading to the end of the track - I could feel the difference. Took it up the road and within 500m I knew it was better.
There was a slight vibration which might have been the road, but the feeling of instability and the mega wobble was gone. I can now adjust my visor without fear of ending up in the ditch

The thing now is to try it out when loaded.
Oh, and get a new tyre pressure gauge!! Lesson learned!!