Why don't you want to store your litecoins on cryptocurrency exchanges? Now it is not clear where it is better to store cryptocurrency, some exchanges provide better protection than many wallets. I can also leave a link to a review of the litecoin core wallet that I used several times.
https://cryptogeek.info/en/wallets/litecoin-core-wallet/chatIt is not safe. On cryptocurrency exchanges, you do not have private keys. Without private keys, you will not be able to fully control your funds.