Transaction fees are not wallet related, in fact they depend upon the miners that confirm your transactions and the number of unconfirmed transactions in the pool. So, even if you switch over to new wallet the fees might remain same. I can recommend you Electrum which allows users to adjust fees and give estimate of how longer it would take for transaction to confirm.
Cold storage wallets are preferred for long term savings but as in the case of OP, he wants to convert a certain amount in fiat of the cryptocoins he would get to fiat and then store the rest in wallet.
Hi Penguincomel, due to decentralization I personally use BitPay and Blockchain wallets.
Blockchain is the underlying mechanism that makes the cryptocurrencies work. I believe you are referring to blockchain.info wallet.
Coinbase does not provide you with your private key and is a centralised wallet, is I would not recommend it. You can earn the referral earnings and move them to your primary wallet like desktop or hardware wallet but it is risky to use third party to control your coins !
In my opinion, you should use a desktop wallet like Electrum and use a trusted exchange to convert the require amount to convert to fiat.