A period, as the following word starts a new sentence.

"I'm hungry," he said.

This is one sentence, containing a full quoted sentence inside the quotes (I'm hungry).

"I want to eat dinner."

This is another sentence. It also contains a full quoted sentence inside the quotes. It is understood from the previous sentence that "he" is still talking, and has started another sentence. That means a period needs to go before this sentence.

It only makes sense to use a comma after "said" if the person you're quoting is continuing a sentence. Here's an example of what that could look like:

"I'm hungry," he said, "but I'm not ready to eat yet."

Here, "but I'm not ready to eat yet." is not a complete sentence. It can't stand on its own, so it needs a comma before it to indicate it continues the previous quoted sentence.

If you want to indicate that the speaker said two thoughts quickly, as one sentence, you should rewrite the quotation so that the pause before "he said" is at the end:

"I'm hungry; I want to eat dinner," he said.

You have several options for dealing with the issue of combining sentences from the end of one paragraph and the beginning of the next into a single quotation.

One obvious option is to format the quotation as a block quote:

The sentence from the end of the earlier paragraph goes here.

And the sentence from the beginning of the next paragraph goes here.

This is formally correct and has the least likelihood of being misunderstood by readers. But it may also bring the flow of your narrative to a standstill; and if you have no other block quotes in your essay or article or book, it will look weird.

Another option is to render the quotation as running text, silently closing the gap between the two paragraphs:

As Sven Yargs has suggested, "The sentence from the end of the earlier paragraph goes here. And the sentence from the beginning of the next paragraph goes here."

But strictly speaking, this is cheating. We've made content from two paragraphs look as though it came from one continuous paragraph. This result may not matter for practical purposes in many settings, but if you are writing in a strict and formal setting—an academic setting, for example, or a legal one—you really shouldn't choose this option.

That brings us to the expedient of using some form of punctuation to signify the removal of the paragraph break. In this area, you actually have several options. One is to use ellipsis points, as Jason Bassford suggests in a comment above:

"The sentence from the end of the earlier paragraph goes here. ... And the sentence from the beginning of the next paragraph goes here."

The drawback of this option is implicit in the generally recognized purpose of ellipsis points. The Chicago Manual of Style, sixteenth edition (2010) lays out this definition of ellipses:

13.48 Ellipses defined. An ellipsis is the omission of a word, phrase, line, paragraph, or more from a quoted passage. ... Chicago style is to indicate such omissions by the use of three spaced periods (but see 13.51 [on combining ellipsis points with periods]) rather than by another device such as asterisks.

Notably missing from Chicago's list of omissible entities for which ellipsis points are a suitable indicator is blank space—specifically line spaces in poetry or paragraph spaces in prose. I would hesitate to use ellipsis points to signify omission of a paragraph break because, in my view, ellipsis points strongly imply the omission of words—not merely of white space—between the quoted sentences; so using ellipsis points as a marker of omitted space might inadvertently mislead readers as to the nature of the omission.

In the case of running together lines of poetry, the established convention for signaling where the end of a line originally fell is to use a slash (also known—with varying shades of meaning—as a forward slash, a solidus, or a virgule) to indicate omission of a line break between consecutive lines of poetry quoted within a paragraph instead of rendered as a block quote:

"I think that I shall never see / A poem jumpy as a flea."

Chicago 16 discusses this at 13.27:

13.27 Run-in poetry quotations. If space or context in the text or in a note requires that two or more lines be run in, the lines are separated by a slash, with one space on either side (in printed works, a thin space to an en space).

[Example:] Andrew Marvell's praise of John Milton, "Thou has not missed one thought that could be fit, / And all that was improper does omit" ("On Paradise Lost"), might well serve as our motto.

But Chicago doesn't address paragraph break omissions at all. The closest it comes to such stronger breaks relates to a situation involving perhaps too strong a break: the whole-line space between stanzas in a poem. Chicago 16 addresses this subject at 13.32:

13.32 Running in more than one stanza of poetry. A quotation that spans more than one stanza should be presented as an extract if at all possible (see 13.23 [on general rules for setting off poetry]). If it must be run in to the text (set off by quotation marks), two slashes (//) should appear between stanzas.

So Chicago offers advice on signifying the omission of end-of-line space between consecutive lines of poetry and on signifying the omission of line spaces between consecutive stanzas of poetry—but nothing on signifying the omission of a paragraph break in prose.

Wikipedia in its lengthy entry on "Slash (punctuation)" suggests using a virgule (slash) to mark a paragraph break:

Less often [than in treatments of multiple lines of poetry], virgules are used in marking paragraph breaks when quoting a prose passage. Some style guides, such as Hart's, prefer to use a pipe ⟨|⟩ in place of the slash to mark these line and paragraph breaks.

The Oxford Guide to Style (2002) strongly endorses using the pipe (or vertical) for this purpose:

5.12.2 Vertical

The vertical rule (|), also called the upright rule or simply the vertical, has specific uses as a technical symbol in specialist subjects, such as computing mathematics, and graphic scansion. ... More commonly, it may be used to indicate the separation of lines where text is run on rather than displayed, for instance for poems, plays, correspondence, libretti, or inscriptions:

[Examples:] The English winter—ending in July | To recommence in August

Christophorus | Codrington | armiger obiit | 1 Aprilis an. dni | 1710 æt. 44

Editors should ensure that this distinction is clear on text. When written lines do not coincide with verse lines it my be necessary to indicate each differently: in such case use a vertical for written lines and a solidus for verse.

Oxford style thus seems to point to this treatment of the sentences in my example:

"The sentence from the end of the earlier paragraph goes here. | And the sentence from the beginning of the next paragraph goes here."

Although use of the pipe is less common in U.S. punctuation than in British punctuation, it is not entirely foreign to me (a lifelong North American). I would be very tempted to use that punctuation mark in the situation you describe, in preference to any other—again assuming that rendering the quotation as a block doesn't work for stylistic or other reasons.