Do I need to run a full node before the 2X fork? There is a meme out there that goes like this: 

"Bitcoin Cash taught people they need to own their own private keys, segwit 2X will teach people they need to run full nodes"

Do I need to run a full node before 2X, or are my coins safe on Trezor?
Your coins are safe on Trezor, you don't need to run full node for coins unless you want to help mine.

If you want to access Segwit2X coin or any other BTC forked coins (in future) asap after the fork, then you should transfer your BTC coin to a lite client and get the private key of that wallet.

Or else, you can wait until Trezor supports Segwit2X coin, in that case, you will get those coins in Trezor itself.