Learn English – I always don’t vs I don’t always

They are not both correct. in searching of is ungrammatical.

Yesterday, I was in search of my keys.

In searching for my keys, I found an old photograph.

In searching for = during, or as a result of, the act of searching for {something}

in search of = trying to find

The right answer is "they always argue"

Because the first part of the sentence "they don't get on well" is present simple" so the following part should be the same. Furthermore, the processing of argument is constantly happening "always", so it implies to something that is not temporary but permanent.