A steering damper, a larger diameter front wheel and skinnier front tire may alleviate a lot of your issues. A spinning wheel wants to stay straight due to the gyro effect it creates as its spinning, the larger the wheel the more gyro effect and the more inherently stable it will be. A skinnier tire will have less of a contact patch and remove a lot of controllability (high speed wobble). It's when we as humans interfere with a spinning wheels gyro effect does it become unstable. As we alter the geometry (rake, offset) the instability becomes MORE or LESS noticeable at speed. You can also change where you sit on the seat, this will change the weight displacement drastically with very little movement forward or aft. One would think that the more forward you are on the seat, the more weight is on the front wheel thus adding controllability and more stability, while this may be true, sometimes LESS is MORE. I've noticed on some of my bikes, less controllability is better so I sit further back on the seat. Another aspect to think about is handlebar placement. If you have handlebars and not forks, try adjusting them forward or aft, this makes a HUGE difference in handling. By moving your handlebars forward you will increase sensitivity, by moving them back, you'll decrease sensitivity to an extent. Another thing to think about if you have handlebars is the handlebar rise, this is how tall the handlebars are. The more rise, the more sensitive the steering feels, less rise, less sensitivity. If you can't change the geometry of the bike, change the geometry of how you ride it, it just might be the solution you're looking for.
-A2M