How do I tell if some oranges I bought have gone bad? (Ok, actually,
they're seedless tangerines, but close enough)

The easiest way to check an orange or tangerine for spoilage is to
sniff it. If it has a slightly fermented smell (like beer or ale), it
has gone bad. Shriveling of the skin is another sign that your fruit
is over the hill.

Open one up and you may find discolouration.