"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 networks. Therefore, to use Kusama, you still need to download the Kusama app from Ledger Live, while your device is plugged in. Once the app is installed, you can follow these steps to import the account into the Polkadot browser extension. Simply choose "Kusama relay chain" instead of "Polkadot relay chain" in the network field. You may notice that you cannot switch networks afterwards; this is normal behavior for Ledger accounts.

You'll also need to change the network in Polkadot-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.