I just went to send some OXT from my wallet to an exchange and noticed that the cost was much higher than if I were to send ETH or UNI while using the same gas price. My wallet shows a different recommended gas limit for each token...is that normal? I thought 21,000 gas was the cost for a standard token transfer (ie. wallet to exchange etc.) on the ethereum network? Is that not the case?
No, 21000 is the minimum transaction cost which is basically valid only for Ether transfers between wallets.
Token transfers can, theoretically, be arbitrarily expensive. A token transfer is a call to a custom token contract which executes custom code. Now, typically the code simply changes the balances of two accounts in its internal ledger and that's basically it, but in some tokens there is extra logic which costs extra gas.
A "typical" token transfer costs around 65000 gas, according to https://etherscan.io/gastracker .