Here is a dvar Torah that explains more about Yehuda fixing up the sin with Yoseph, and it also continues on the theme of miyadi tivakshenu, from my hand you will ask of him.
Reuvain also wanted to accept responsability for Binyamin, but Yaakov did not listen to him. Let’s look at the difference between Reuvain and Yehuda. Reuvain placed the bad result (my two sons shall die if I dont bring him back) before taking responsability (give him into my hand and I will return him). Yehuda, on the other hand, said the responsability (I will guarantee him from my hand you will ask him) before the result (that he would lose his olam habba). This shows that Yehuda was more interested in the responsability, while reuvain was more interested in avoiding the bad result. Who would you trust more?
Secondly, the gemarah says that Reuvain’s words of responsability are kablanus. A kablan must pay up even if the lender has money. This means that he does not work with the lender to see if he has money or not. This means that Reuvain was detached from Binyamin. He would protect Binyamin only because he would protect himself from losing his children. This did not fix the sin of hating Yoseph, so Yaakov did not trust him.
Yehuda on the other hand said areivus, and Yisrael are similarly areivim ze laze. An areiv pays only if the lender does not have money. Yehuda would work with Binyamin. If Binyamin could protect himself, Yehuda would allow it. If not, Yehuda would protect him. By working with Binyamin, this showed that he loved him, not only himself. Who would you trust?
Now here is the amazing part.(tell me if you flip out)
Yaakov said to lavan, a trayfa I did not bring you, I would make up its loss (anochi achatena) miyadi tivakshena. The trayfa is Yoseph, as it states, tarof taraf Yoseph. Yaakov (propheticly) said that he would make sure that the loss of Yoseph would be fixed up (anochi achatena), and that he would be responsible for it (miyadi tivakshena).
Now Yaakov simply had to wait for the son who said the same thing. Yehuda copied his theme and his words. Yaakov said anochi achatena miyadi tivakshena, and Yehuda said anochi e’ervenu miyadi tivakshenu. When Yehuda said anochi e’ervenu this is the same as anochi achatena, because the areivus (of loving Binyamin) fixed up the sin of hating Yoseph. And the responsability of miyadi tivakshenu is the same as Yehuda’s miyadi tivakshena. Because Yehuda said the same thing, Yaakov trusted him.
Moreover, the one that would fix the sin of Yoseph (anochi achatena) would thereby be worthy of Dovid Hamelech and moshiach. Yadi (my hand) is Dovid (see before)and tivakshena is moshiach that we are mevakesh. Yehuda said anochi e’ervenu and fixed the sin of Yoseph so he merited miyadi tivakshebnu, Dovid and Moshiach.
Moreover, as I stated before, Dovid, yadi, would also be asked (tivakshenu) of Binyamin in regard to shimi ben gera and Mordechai. And ?????? alludes to ?? ???, and Mordechai is called ben yair, ben shimi (ben gera), ben kish, ish yimini.