Auto meshing doesn't ensure adequate connectivity between member to member and is therefore recommended for horizontal area elements enclosed by line elements only (floors), where there is no structural connection between floor and any element in between the panel.
In case of vertical elements, connectivity between vertical and horizontal elements is of due importance and is better to be achieved through manual meshing.
In case of auto meshing as you indicated, change in size of auto mesh could solve the problem as its possible that connections inadequate (nodes not coinciding) on 1m element size can be adequate for 1.2m size (nodes start coinciding) and so on.
For p-delta,a possibility exists that due to any meshing error some connection is modelled with inadequate lateral stiffness i.e when program try to impose lateral deflection due to seismic loads,modes start yielding frequency below shift.
If you are interested in studying the problem,then easy approach is to check each mode shape and investigate the member that is going in unrealistically large displacement. Solving this member's connectivity inadequacy will solve your problem.