My head bearings were nearly rusted solid at 12000km and there was no movement in the forks and jacking it up and swinging the steering felt smooth and predictable.
Riding the bike was EXACTLY as you described...falling into corners and having to quickly correct mid corner to lift it back up!!
I thought it was tyres and changing pressures etc made a small difference but then I bit the bullet and dropped the steering stem down for a loom and was blown away at the rust!!

It could be tyres although my Tourances were awesome and always felt stable and predictable till I changed them out to TKC 80, but whatever the age...check and grease the steering bearings!
Goodluck
Matt