HFSS is really bad at meshing curves. Avoid any automatic curve difinitions. Instead use extrusions with multi-line patches, or at least stepped function curves, where you can set up number of steps explicitly. The same is for circles and cylinders. To avoid false "numeric" resonances - rotate your circular objects to make them "out of phase", i.e. no parallel faces and periodic edges.
- - - Updated - - -
Moreover, when examining a spiral, don't forget that the nearfield can be very complex, which HFSS does not expect when refining tetrahedra. Therefore inside the airbox create another small airbox around the spiral with dense "initial mesh inside" property.
- - - Updated - - -
BTW. Your spiral is made of air. Maybe that's an answer?