Airbags are normally run in joined pairs. They are designed for one of two uses... Comfort or load-assistance. None are designed for anti-sway. They will actually add sway, in most cases. If you get airbags installed, get the ones for load-assistance (rigidity), not comfort (bounce). Also, if available, get the ones that auto-level and are independent controlled. That helps with side-loaded cargo issues.
Springs are a raw "last resort", which will just bounce you around, depending on your load-size. You need shocks to dampen the bounce, but if your load is not balanced, you will still sway. You will just have a more rigid "twerk", instead of a soft "twerk". "Jump around, jump around, everybody get up and jump around!"
First check that your load is balanced. Even if nothing has changed, it may have just always been "off". Now it is seeing the damage from having a high side-loaded vehicle.
If you are "twerking"... shifting side to side, or teetering side to side, you may need an anti-sway-bar, or need to replace one. (Not all vehicles have one.)
They help stop the vehicles left/right pairs from moving independently, which causes that "twerking" motion. Both go up and both go down, within a determined range-limit.
Unless you are a tank or a jeep, intending to do rock-crawling with four-wheel-drive posi-traction, then you want anti-sway-bars. You also want to make sure that they are functioning.
Where they fail... The mountings can lose the rubber mountings. The metal bar can "twist", so one tire is always with more pressure UP, with the other mostly DOWN. That makes you twerk on every bump, as it throws your car to the side, when any wheel is hit. (A low shock/spring/leaf/air-bag will also do that too, with a normal anti-sway-bar. But you will see one tire has a lower gap in the wheel-well, which may also be a loading issue.)
Always replace springs/shocks/leaf-springs in pairs. Air-bags are not that much of an issue, as they are usually joined loads or independent balanced units. If it is your shocks, springs or leafs... Think about upgrading to a more rigid set. The "normal set" that you have installed is obviously not suited for your constant load. Most are sold to sustain a comfort ride, unloaded, with temporary "high-load" limits. You essentially are rebuilding for a higher tonnage vehicle now. If it was a 1-ton vehicle, you now need to replace things as if it were a 2-ton vehicle. If you were a 4-ton vehicle, the extra ton of "stuff" in the vehicle will not matter as much. That is only a 1/5 new-load going from 4 to 5, not a 2x new load, going from 1 to 2.