let me try the first fragment - as part of the arrangement, he received severance.
I believe as and like both can bring suspicion to this fragment - in one case the preposition as suggests a likelihood of the arrangement part-then we need to use like, and yet the other instance could be as is used as the actual part of arrangement prescribed-then we need to keep as

His anxiety about his company's future is ill-founded. - we have got two his here...

His anxiety, that his company may be sold, is ill-founded. - I figured missing commas here ... also anxiety couldn't be ill-founded, because anxiety is a sort of feeling and this can't be measured with any foundation.

OneTwoThreeFour wrote:According to MGMAT, the sentence below is both right and suspect:

as part of the arrangement, he received severance.

Why is that?

In addition the statement below is right,

His anxiety about his company's future is ill-founded.

But the statement below is wrong:

His anxiety that his company may be sold is ill-founded.

Is the reason "about" is wrong in the second sentence because "about" must be followed by a noun phrase?

Thanks!

For the first sentence, "He received severance" is already a complete sentence. "As part of the arrangement..." is a modifying phrase describing the conditions under which he received severance.

You could also switch the clauses and the sentence would still make sense. He received severance as part of the arrangement.

A few more examples like this:

As per the contract, she moved out on the first of the month.

In accordance with her wishes, the son buried his mother in the family plot.

As part of the bet, he cleaned my room for me.

For the second sentence, it is idiomatically correct to say you have anxiety ABOUT something.

I am anxious about my GMAT score.

She's anxious about her wedding.

The soldier's wife was anxious about her husband's deployment.

The "anxiety that" sentence is awkward and wordy. You could say simply, "His anxiety is ill-founded." or "His anxiety about his company is ill-founded."

You are correct that when we use "anxious about" we need a noun to follow to indicate who or what is creating the anxiety.