the klr is tall.... that ok with you?
wobblyness in the suspension is largely due to improper servicing of the forks (especially) and the rear shock sag setup. the fork oil needs to be measured with the springs removed and the fork collapsed... there is a distance to measure down to the oil from the top of the fork. distance depends on Generation.... older needs 170mm, newer needs 135. Ricor valves make it better.... 5 wt oil is set to 40mm. I drive mine at 75 all day (16 tooth sprocket)... no wobble
tire choices dictate how they hold the road... same as any other bike. the klr can be driven down right sporty no problem
08 and early 09s have had problems with piston rings that can make them burn oil. the fix is a piston replacement. there are aftermarket kits to make them 685, 688, and 705cc. all are better than stock.
the "doohicky" is a problem. even the Gen II bikes , though not as bad.
theres more.... still interested?