The only thing you have to "relearn" is that a full-stop at the end of reported speech becomes a comma when the sense runs on into the sentence which reports it.
"I'm done," she said.
That's one complete sentence reporting some speech.
It doesn't matter if what is being reported has its own sentence structure: that's contained within the quotation. It's the comma at the end which links the quotation to the reporting. Please be reassured that your example is correct:
"I can't take her. I'm done," she said.
Where you're reporting a trailing-off of speech, or some other ending, the punctuation mark isn't a full-stop so it doesn't become a comma. However, because the sense runs on into the reporting clause, that doesn't get a capital letter, in much the same way as the first example.
"But I thought he was..." he complained, lapsing into silence.
"Oh sh—" he whispered, seeing the car hurtle towards him.
"No!" she exclaimed, slamming the book shut.
Where an interjection interrupts the reported speech, I'd use a dash. There's no punctuation at all at the end of the first fragment, because there's nothing there in what's actually being said.
"I just can't" — she sobbed ostentatiously — "take any more!"
As Robusto points out in comments beneath the question, there is no universally acknowledged rule governing whether to include or omit a comma after a conjunction at the beginning of a sentence. Robusto reports preferring to include such commas in academic documents, but many other writers and editors would not include them.
In my experience copyediting manuscripts for book publishers (including university presses) and later for magazine publishers, I don't recall ever having encountered a house style that required adding a comma after "And," "But," or the like. To the contrary, most house styles either said nothing at all on the subject or recommended omitting such commas, presumably for the reason that Words Into Type, third edition (1984) gives at the start of its long section on comma usage:
A comma should be used only if it makes the meaning clearer or enables the reader to grasp the relation of parts more quickly. Intruded commas are worse than omitted ones, but keep in mind at all times that the primary purpose of the comma is to prevent misreading.
The argument for including a comma after an opening conjunction is not, I think, grounded in a desire to make the meaning clearer (since the meaning tends to be quite clear without the comma, as Peter Shor indicates in a comment above), but rather in a desire to demarcate with exactitude the boundaries of the parenthetical expression that follows. Why Gregg Reference Manual would insist on such precision at the beginning of a sentence but not in the middle of one is a mystery to me.
There is nothing inherently wrong with using commas to break out parenthetical phrases regardless of where they appear in a sentence: It increases the number of commas in a work while (arguably) not making the sense of the text any clearer; but it's a style decision, and style decisions—if followed consistently—don't need to be justified.
On the other hand, if you don't want to add a comma after a conjunction at the start of a sentence, I don't think that you should consider yourself to be under any obligation to the preferences of Gregg Reference Manual unless your publisher has instructed you to obey it.
Best Answer
I don't think that any settled and widely accepted rule for handling "namely" exists, which makes the challenge to "prove your claims by including references" rather difficult. The link you provide to Merriam-Webster's Learner's Dictionary is useful because it describes some of the numerous ways that people deal with "namely," including these:
But that source doesn't account for other possibilities, such as these:
As the contrasting comments and responses to your question indicate, people have very different preferences for punctuating "namely." I favor the —namely, option, which I like because it establishes a clean break from the wording responsible for the deficiency in identification that the words following "namely" are dedicated to supplying. Nevertheless, I can't claim that my preference is anything more than a preference.