Yes, in fact ‘quod’ is just about the only thing correct in these sentences. But don’t be discouraged. Mistakes are how we learn.
In your first sentence re-check your principal parts for your verb. i is the correct ending, but the stem is wrong. Secondly, you need to say ‘I was tired’ not ‘he was tired’ as you have.
In your second sentence ‘tardus’ does not work because the ‘us’ ending is describing something masculine, and ‘it’ is not masculine. It would be better to ue an adverb like ‘tardius’ = ‘it was rather late.’
Again in your third sentence you need a different verb ending from ‘erat’. You want to say ‘I was hungry’. This is not great way to say that in Latin, Try to find a verb that means ‘to hunger’ and use it instead.
Others may be wiser, Scribo, but I don’t see a distinction between “quod” and “quia” and “quo” (or “quoniam” or “quando” explaining from the writer’s/speaker’s sole viewpoint) in replying to “cur”. I don’t know any rule to the contrary about that (which isn’t saying much). Alii me sapientiores sint, Scribo, at distinctionem exstare inter “quod” et “quia” et “quo” (et “quoniam” et “quando”, sub judicio oratoris/scriptoris solo) ad “cur” in respondendo non cogito. De hâc re, ullam regulam quae prohibet ignoro (quod non multum significat).
Cur discedis?
Quia/quod/quoniam/quando sero est [at hîc non “quo” ut sentio]
Et quia/quod/quo/quando/quoniam debeo iter facere.