You’re right to be concerned, because that "shimmy" is usually the first stage of Death Wobble. In these newer RAMs, the biggest weak link is often the track bar bushings or a loose track bar bolt. If there is even a millimeter of play there, it will allow the axle to oscillate. You should also check your drag link and tie rod ends for any premature wear.
To answer your question about the steering stabilizer: Yes, it is 100% a band-aid. A stabilizer’s job is to damp vibrations, not to hold the steering geometry together. If your front end is tight, the truck shouldn't wobble even without a stabilizer.
A dual-stabilizer setup is a solid upgrade for overall steering feel and to prolong the life of your steering components—especially if you've gone to larger tires—but it won't "fix" a mechanical issue. Start by having someone dry-steer the truck (turning the wheel back and forth while you're under it) to see what's actually moving. If you want a real fix, look into a heavy-duty adjustable track bar and a steering box brace.
