Sometimes there is just no substitute for nominal trail specs, and yours seems to be one of the trikes that exemplify that. With its stock rake of 34 degrees the need for trail correction will manifest itself just as you have described. Sometimes I make assumptions, so if you aren't familiar with trail a bit of searching within this forum will bring you up to speed. Reducing the trail measurement is exactly what the 'raked triple trees', or 'EZ Steer kits', or similar solutions are directed toward.
If you plan to stick with the trike for the foreseeable future, invest in a good raked triple trees kit. They can be a do-it-yourself one afternoon project if you are mechanically inclined. For your softtail my personal opinion would be 4-1/2 to 6 degree trees, although a professional supplier (i.e., Randy, aka Bazooka on this forum - highly recommended) could be more specific with you.
One more fix you might try before you pop the bucks for the steering kit is to tighten your current forks' stem nut a bit, or at a minimum make sure it is up to the maximum recommended specs. That is a critical point for trike steering and if even slightly loose (or not tight to specs) can be the culprit to exactly the behavior you describe.
Good luck, I feel quite certain that with a little tweaking in this area you can stop your wobbles, and the trike will become your best friend.