Does improper alignment cause permament damage ?

Depends. If you have a low tracking force(2.5gm or less) , a clean LP and just a minor error, I doubt it would matter. A worn stylus is gonna do more damage.
I mean with an eyeball setup with the cartride, 'near' (within a mm or 2mm) of proper overhang is just not gonna damage LPs.
I KNOW I will invite getting caned by saying it, but the myth of needing perfect cart alignment is not true.

From what I learned from various sources in the industry, yes. Poor tracking bad enough to cause a lot of distortion sounds like something that should be addressed.Since you listen to these records they must have enjoyable value.The stylus and record both will suffer from extra wear.A damaged stylus will ruin a record also.I would want to correct the problem since it may not be to late.I hope you caught it in time.

As Elizabeth says, it depends.

1. Does this distortion happen in synch with dynamic peaks in the music or is it more random?

2. Can you describe how it sounds?