A genitive absolute is only used when the subject of the genitive absolute has no grammatical relation to the rest of the sentence. So in your first example you cannot use a genitive absolute because the subject in both parts of the sentence are the same. (in greek you just could use a participle, eg. I, being hungry, will prepare food)
Just keep in mind that the subject of the genitive absolute must be different from anything else in the sentence, ie. subject, object or indirect object. Completely grammatically unrelated. Your second sentence is a very good example of this: Achilles has no grammatical relation to the dying men.
“Cum ego verbo utar,” Humpty Dumpty dixit voce contempta, “indicat illud quod optem – nec plus nec minus.”
“Est tamen rogatio” dixit Alice, “an efficere verba tot res indicare possis.”
“Rogatio est, “Humpty Dumpty responsit, “quae fiat magister – id cunctum est.”