commas - Is this kind of syntax correct?

I wonder if a sentence is still correct if you use an introductory adverb as well as an introductory phrase in front of it? If this is the case would you have to use two commas?

Moreover, according to the text, Thomas is the one who breaks down
after the infanticide took place and his heartbreaking cries resound
through the house and even the neighborhood (6).

It sounds weird to me...

There is another reason why I find his sentence strange: it kills the reading flow of the sentence. If you used parentheses, it would sound much better from my point of view, but I don't know if that is a wrong thing to do because I have never seen it before in writing: "Moreover, (according to the text), Thomas is the one..."

Revised version after discussing this in the comments:

Moreover,(according to the text) Thomas is the one who breaks down after the infanticide has taken place, and his heartbreaking cries resound through the house and even the neighborhood.

or

According to the text, moreover, Thomas is the one who breaks down after the infanticide has taken place, and his heartbreaking cries resound through the house and even the neighborhood.