Grammar Question

'...I said with a little more enthusiasm this time.' You could go either way. The comma does not change the meaning.

'...I said, running along the street.' I think you necessarily need the comma there. Really, you are saying a short form of '...I said as I was running along the street.'

'...I said as a feeling of dread washed over me.' Either way, the comma does not change the meaning.

'... I said before Frank could open his mouth.' I would put the comma there. Think "I said 'before Frank could open his mouth.'"

ParchmentPaper said:

Grateful for the reply Al X Thanks. What I could really do with is a reference to the rules that would apply to this type of situation, something I have sadly failed to find so far

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https://owl.english.purdue.edu/owl/owlprint/607/

In simplistic terms as it relates to your examples, think of it this way. You have two independent clauses. If the presence of a comma between them changes the meaning, you either need it, or need to omit it depending on your intended meaning. Otherwise, it's a style call. It gets a little more complicated than that and the above is a fairly good breakdown. You just have to read through it to find an instance applicable to your situation.

'I helped my uncle Jack off a horse.' Commonly cited example but the capitalization of Jack indicates a proper name and a comma is actually not needed. Wouldn't hurt though.

'I helped my uncle jack off a horse.' While you may well be assisting your uncle in his horse breeding business, a comma won't do anything here other than make the sentence awkward. Not that is isn't already.

I think you're worrying over restrictive and non-restrictive content, the difference between:

The swimmer felt the waves crashing into him.
V
The swimmer walked the beach, stopping here and there to pick up shells.

In the fist, 'crashing' is bringing in adjective phrase and modifies waves (restrictive). In the second, you have a participle phrase (stopping here and there...) that modifies the swimmer, but is only loosely tied to the main clause and added information (non-restrictive). If you don't have the comma in the last example, there's potential for ambiguity over the beach being the one to stop and pick up the shells

The swimmer walked the beach stopping here and there to pick up shells.

Using the comma helps to clarify it's additional info that relates back to the main subject. Although you can change this around:

Stopping here and there to pick up shells, the swimmer walked the beach.

If you change the other one around, there's potential ambiguity over the swimmer crashing into something.

Crashing into him, the swimmer felt the waves.

So it can be down to who/what is being modified at the time.

I said with a little more enthusiasm this time.

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I wouldn't. But there are stylistic choices allowed here.

I said, running along the street.

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Lol, welcome to English and the it can be an adjective phrase v participle here. I said running along the street, v I said, running along the street. I'm tempted to use the comma here.

I said as a feeling of dread washed over me.
I said before Frank could open his mouth.

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I wouldn't use a comma. these feel restrictive.

Is the absence of a comma after 'I said' correct and why?

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I'd consider whether you're tempted to add the comma more because you're instinct is to pause sometimes. That's not the case with writing. The commas are used to show a loose tie and break the sentence down into it's parts. E.g.,

"The tide's in, Joe." Vocative

Here Joe isn't a part of the main clause, the comma shows he's addressed directly, but still only loosely tied to the main clause itself.

But if you look up restrictive v nonrestrictive content (with prepositional phrases, participle etc) it might help you a little more.

Have someone sight-read the dialogue and tags aloud, and see if they frown or hesitate while doing so - hesitation often means there's a lack of clarity/ flow that needs attention.

Most good sight-readers can see far enough ahead to comprehend what they are reading.

ParchmentPaper said:

I said, running along the street.

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That's the only one that requires a comma, and it's because of that modifier "running along the street" at the end. Those are always set off by commas. All your other examples are fine without the comma.

Purdue Owl's Quick Guide to Commas. See rule #8.