A syntactic parsing of the sentences could help to show what is going on. The word "that" in your examples is a marker of clausal subordination. That is, the word "that" has no semantic meaning.
This is a way to parse your examples:
1.) He said [that, after considerable contemplation, he would retire].
2.) She said [that, in 1969, she and her husband went to Woodstock].
3.) Please be advised [that, on this day in 1954, a resolution was achieved].
Notice how it so happens that in your 3 examples, that the word "that" is necessary to prevent ambiguity or mis-parsing by the reader. Look at what happens when the word "that" is removed:
1.b) He said, after considerable contemplation, he would retire.
2.b) She said, in 1969, she and her husband went to Woodstock.
3.b) Please be advised, on this day in 1954, a resolution was achieved.
Notice how the meanings have now changed: the expression delimited by the commas are now most likely being interpreted as being part of the matrix clause. That is, for #1.b, after he had contemplated for some time, he then said something about retiring. For #2.b, in was in 1969 when she said something. For #3.c, today is a day in 1954 (while the original #3 was saying that 1954 was way in the past, e.g. December 7, 1954 while today is December 7, 2014).
Notice that the way the comma pairs had been inserted in your three original examples is one clear way of making sure the sentences are parsed and interpreted correctly. (If one or other of the commas is removed, then the sentence might be misinterpreted.)
But some writers, especially for fiction prose, might do their comma punctuation differently. For instance, it is possible that one of the commas might be omitted (for a lighter punctuation style) in similar sentences. But care needs to be used to make sure that the word "that" doesn't then accidentally become a word with meaning (instead of a mere marker), and to make sure that the result doesn't create a sentence with a different meaning from the original.
Summary: In your examples, the word "that" is a marker of clausal subordination. In your examples, they are markers of the beginning of a declarative content clause. The pairs of commas help the reader to parse the expressions within the subordinate declarative content clauses correctly.