I suggest 'us' going on an excursion this Sunday...is wrong! I suggest we go on an excursion this Sunday. I suggested our going on the excursion last Sunday.

No, I would not use it in this context. “I suggest ‘we’ go on an excursion this Sunday.” I would write it this way. "We" is the correct pronoun in this sentence. "I suggest" is (the main clause) the present tense. "We" is the subject of the verb "suggest." "Go" is the verb. "On an excursion" is a modifier (also known as an adverb of place). "This Sunday" is an adverb of time. We use the pronoun "we" as the subject of a verb. "Us" is the object form of "we." I would not use "us" in your example – unnatural to me; nevertheless, I would use it in this context: “They invited ‘us’ to go on an excursion this Sunday.” We would not say, “They invited we.” Moreover, here we are using the pronoun "us" as the object of the verb "invited." "Our" is the possessive form of "we." Furthermore, it would be incorrect to use "our" in your example. However, you can change the phrase "going on an excursion" into a compound adjective. It would mean that you must hyphenate it. “I suggest our ‘going-on-an-excursion’ trip this Sunday.” "Going-on" is a compound adjective, which is modifying the noun "trip." Use "our" when it means belonging to or connected with us. Other examples: Our excursion will begin with a bicycle ride this Sunday. Other examples: Our dog/cat/book/toy/mom, etc.

You can use either one in this particular sentence, but this construct of words is a challenge. She saw us going on an excursion this Sunday. Our going on an excursion this Sunday was the major topic of conversation on Monday. I want to think about us (or our) going on an excursion this Sunday.