"I remember watching Dallas," vs. "I remember us watching Dallas."

Question

I remember watching Dallas.
I remember us watching Dallas.

I know that both are common usage. What I don't understand is exactly what role "watching" has in the second sentence. It is clearly a gerund and direct object of "remember" in the first. In the second, "us" is clearly an object, as it always is. I think most people will tell me that "watching" is then an adjective or adverb ("I saw John walking down the street"), but this doesn't sit right with me, because the act of watching is still the main thing, and "us" is actually only providing more detail about it. But, grammatically, "us" and "watching" cannot both be nouns.

Some will say it is an adverbial phrase, but that should have a comma: "I remember us, watching Dallas." It just seems unlikely to be using "watching" there in a completely different way than when "us" is not there.

Another example:

We must address the issue of children using mobile devices in class.

The issue is not so much children as it is the usage.

"It ended with me calling you to find out the security code."
"How do you feel about Putin winning another term?"
"One wrong move could mean you going to jail."
"This is me walking out."

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Edit: The answers to the question linked as a possible duplicate treat the word "him" as the subject of a clause, which it categorically cannot be. This is incorrect and an improper understanding of the grammar of the sentence.

To further illustrate, the kind of phrase I'm talking about can be used as a subject:

Children using mobile devices in school is becoming a serious problem.

Now, would you replace "children" with "they," as you would have to if it were the subject?
They using mobile devices in school is becoming a serious problem.
NO. That would be incorrect. So "Children using mobile devices in school" is not a clause, as claimed in the answers to the other questions, since a clause requires a subject and predicate.

Others would say:
Their using mobile devices in school is becoming a serious problem.
But, putting children back in, this would mean:
Children's using mobile devices in school is becoming a serious problem.
And "using" here cannot be a deverbal noun as it has a direct object. I would say:
Them using mobile devices in school is becoming a serious problem. OR
Me marrying you was the best thing that could have happened to you.

Answer 1

I don't know the technical term for it but I call these noun phrases. The entire phrase functions as a noun.

Watching is not an adjective or adverb, it's always a verb. But it makes up part of a noun phrase in the second sentence. I've surrounded all the noun phrases in parentheses below and provided alternative phrasings in square brackets afterwards:

I remember (us watching Dallas). [the time we watched Dallas]

I saw (John walking down the street). [John's march]

We must address (the issue of children using mobile devices in class). [This one is fine, I can't really make it clearer]

It ended with (me calling you to find out the security code). [my call to you about the security code]

How do you feel about (Putin winning another term)? [Putin's reelection]

One wrong move could mean (you going to jail). [your gaoling]

This is (me walking out). [my walkout]

You can replace the phrase in parentheses with the phrase in square brackets and the meaning is retained but it's clearer that the highlighted phrase is a noun phrase.