'C' bars will give you the same effect as risers, probably a couple of inches in height at most. Trouble is, second-hand 'C' bars are getting harder to find as it's a popular modification (the only additional change needed is the master cylinder to steering tube brake hose if the bike is an older one).
The weave could be the tyres. It could equally be worn or poorly adjusted steering bearings. There's also the possibility that the fluidbloc steering damper isn't damping.
Replacement of the bearings/damper is cheaper than a pair of tyres. Adjustment is even cheaper!
If the bike is in need of tyres anyway, I'd do that then see if the steering needs attention.
These bikes should track straight at any speed they'll reach and be stable. Where they suffer by comparison with more modern bikes is the tyre choice, no-one makes radial rubber for them in the standard sizes any more. Bias belted are good enough, they just don't like over-banding and studs in the wet, they'll skip off such things and if you are used to radial tyres it may surprise you. The effect feels far worse than it actually is.