The Masorah Katanah states that in five places are there words which appear with these two ta'amim.
And the Masorah Gedolah goes on to explain about the Mesorah Katanah, that these two ta'amim in a single word are Telishah Gedolah and Garish. It says the marker for where they appear is זה, קרבו, שובו, ולאלה, זאת. This points to Bereshit 5:29, VaYikra 10:4, Melachim-2 17:13, Yechezkel 48:10 and Tzephaniah 2:15 as clarified by the Komarna Rebbe. It goes on to explain that the reader pronounces the Garish before the Telishah even though the Telishah Gedolah appears first, at the beginning the word. It states the general rule is that Garishin are first and Telishah is afterward. It then explains that marker for this rule is taken from the posuk (Devarim 26:12 כי תכלה לעשר) discussing the requirement of giving ma'aser from the field in the third year to those who have no inheritance in the land (like the idea of גרושין) before benefitting from the harvest of the produce (like תלשא גדולה). And this follows the explanation of the Komarna Rebbe in his commentary to the Masorah on Bereshit 5:29 called Menorat Shlomo.
Telishah Gedolah can have two possible meanings. One is like a big harvest, תלש means to pick, or to pluck, or to detach. But it can also have a connotation of harvesting kindness. גדולה can also be understood to mean kindness. Among Sephardim this trope is called Tirtzah which has a connotation of pardon and reconciliation. That would seem to imply the idea of harvesting kindness is more appropriate.
Gershayim has a connotation of divorce, or separation, or banishment. Since it is plural it means that it relates to two or more. Again, the Sephardic name is Shnei Gerishin, meaning two levels of separation.
In terms of trope, in general, according to kabbalistic understanding, they pertain to the Heavenly influence that is actually driving what is occurring below in the physical world.
Additionally, there is a concept of Kri v'Kativ, where what is written is not how something is pronounced. This has a connotation that how something appears below in the physical (the Kri) is different or opposite from what it is above (the Kativ).
In the context of your first quotation, the dual trope appears over the word "This (one)" זה, meaning Noach, will give us comfort from our acts and from the pain and toil of our hands in regard to the soil which G-d cursed.
This is referring to two ideas. The first mentioned is our acts. This is the sin of idol worship which began in the generation of Enosh ben Shet. There were 8 generations from Enosh to Noach but the generation of Chanoch was not included in the count because they did teshuva with Chanoch's help.
The second idea is the pain and toil of our hands which relates to the one of the consequences of the sin of the Tree of Knowledge. The soil of the earth was cursed and the quality of the produce decreased. They would plant wheat and thorns and brambles would sprout.
Noach was going to be instrumental in removing the consequences of these two sins from the world. That would manifest in the physical as the flood. But the hidden, Heavenly influence driving all of this was actually great kindness and reconciliation.
The 7th generation after Enosh would remove the judgement (the fruit of the five Gevurot) from the world. And this is alluded to in the word which carries the double trope (זה).
In the citation in VaYikra from parshat Shemini 10:4, there seems to be a similar theme. There are two brothers involved in improper behavior, Nadav and Avihu.
But even though what happens to them appears physically and externally to be the consequence of sin, the commentaries emphasize according to Moshe's words to his brother Aharon (בקרבי אקדש) that their offering was found favorable and accepted (תרצה). Their remains were treated just like any other offering on the altar.
The problem with their activity was because it wasn't commanded by G-d. In the language of kabbala, it was Ratzu (approaching HaShem) without Shuv (drawing that influence back into the physical world). The two brothers were detached (תלישא, גרשים) from drawing HaShem's blessing (גדולה) back into the physical world. And this is alluded to through the word b'Krovai (בקרבי) being spelled without the letter Vav. That Vav hints to the six behaviors lacking in the actions of the two brothers enumerated by the Tur to VaYikra 10:2 and that this lacking results in detachment from the physical world.
And this is alluded to in the word which carries the double trope Kirvu קרבו, which can be understood both as a command to Mishael and Eltzafon to approach the remains of their brothers and also to the fact that Nadav and Avihu approached (without returning).