In your experience, how can the axis of the head be adjusted? My fear is the band inside the cutting head that twists the blade housing has somehow gotten loose and now isn’t “straight”.
The timing belt in the cutting head needs to be somewhat loose. (remove the cover of one of your other machines, it should be similar)
As long as it doesn't jump teeth, you are ok.(I don't think it does, since you say the cut is fine in the beginning, then at some point gets bad and is good again when you start a new job)
I think, i just found the possible cause of your problem. It must be the encoder in the tangetial motor, or the cabling, or the counting electronic circuit.
During initialization, the blade turns and stops at the "home" signal of the encoder. Then the offset is added, so the blade looks in the correct direction. During cutting, the mainboard only looks at the AB signals to keep the angle correct. If there is a problem with these signals, the blade angle gets wrong and you get your cutting results.
Now the problem could be everywhere from the encoder on the motor, the cable to the head board, the head board, the flat cable going all the way to the mainboard and the mainboard itself. Id reseat all these cables first.(You don't need parts for that)
My next step, test it with a known working head.(highest likelyness)
Then the flat cable from the head to the mainboard could have a defect. (Happens in heavily used cutters, but rarely)
Then the mainboard.( rather unlikely, but possible)