The majority of exchange sites are giving you daily limit even when your account is not yet verified and you can still convert your money. But, i prefer kucoin or binance if you are looking for a cheap website to convert your bitcoin to the ethereum. Sometimes im also using a crap exchange site crex24 too caused by it will chart you a little fee and there is no limitation for how much money that can be withdrawn by the new account.
this based on my experience.