I have been using a Warco Economy Mill/Drill for about 20 years. Yes, it is not ABSLUTELY rigid, and the head needs to be clamped to the column quite hard. If it is not, a cutter will dig in and shift everything.
It sounds as if your machine has play in it, possibly in more than one place. (I am assuming that your cutters are sharp; and that you are not climb milling )
My advice is to go through the machine check for clearances and eliminate them. The alignment strip is only an approximation, from what you say.
Is the column firmly fixed to the base?
Are the X and Y gibs correctly adjusted?
Is there a lot of play between the quill and the head?
Are the quill bearings, in the head in good condition, and correctly adjusted. They may be taper roller bearings. The manual should show you, and possibly advise on how tom preload them.
In any case, the head needs to be clamped firmly to the column
As you eliminate unnecessary clearances, things should improve.
HTH
Howard
Edited By Howard Lewis on 02/07/2019 20:23:28