If Carpathia was not full when she rescued Titanic survivors, why was she so overloaded? There should have been plenty of room and spare cabins.
Indeed, Carpathia had a lot of spare space. The problem was, most of that space was steerage.
After a refit in 1905, Carpathia could carry 2450 people, according to the statement given by captain Ronson to the US Senate Inquiry commission, 250 in first and second class combined, and 2200 in third class dormitories. The captain might've been off, though - he states several times that he does not know the exact capacity of his ship (it seems weird to me, but I'm not that well versed in ship operation; also the captain might've tried to dodge questions related to the problem of ships of that time having rescue equipment vastly inadequate to the number of passengers carried). Wikipedia lists a bit different numbers, although with no source provided: 100 first-class, 200 second-class, 2250 third-class for total capacity of 2550. Thus, if we go by sheer capacity - even with lower capacity cited by the captain, Carpathia could've taken in another thousand passengers.
I couldn't find the exact number of first and second class passengers onboard the Carpathia during the Titanic disaster. Data given here lists ~130 first class passengers (which does not agree with wiki 100 first-class passengers, but still - the first class wasn't empty). Carpatia rescued 201 1st class passengers, 118 2nd class passengers from Titanic. Carpathia's crew might've been reluctant to put first and second class passengers in steerage - and that would lead to ovecrowding in those classes.