"I noticed there's a separate Ledger app for Kusama. Do I need both apps for managing DOT and KSM, or can I just use the Polkadot app for both tokens and simply switch the address format on Polkadot.js between Polkadot and Kusama?"

Polkadot and Kusama are completely separate. Therefore you do need to install both apps on your Ledger, and you also need to change the network on polkatot.js, depending on which one you want to connect to. 

Here you can see a short video, which details how to add your Ledger Kusama app to polkadot.js and how to send funds from it. (Please note this still shows the old layout of polkadot.js - the menu moved from the left panel to the top in the meantime.)