Is there any GUI wallet that lets you manually set the gas price? I've tried MyEtherWallet and Mist but both only have a sliding fee bar with an upper limit to the fee. Is my only alternative a command line wallet like Geth?
The Parity UI lets you set the gas price. When the "Signer" pops up the transaction confirmation dialog, there is blue text that reads "EDIT CONDITIONS/GAS/GASPRICE". Click this blue text and a new dialog lets you change the gas limit and gas price. The default is 20 Gwei (2 followed by 10 zeros). These days 2 Gwei is enough but slow. See EthGasStation.info.
While the MetaMask chrome extension isn't a wallet per se, it does allow you to set the gas price. This is useful if you want to control the gass price that a Dapp uses for its operations. The transaction confirmation dialog includes a field you can edit.
As of today, neither MyEtherWallet nor MetaMask truly allow you to set the gas price. On neither wallet will you be able to actually send the maximum amount of Ethereum in your wallet, due to their built-in calculations.
The slider mentioned above for MEW doesn't exist any longer, and MetaMask doesn't allow you to click "next" when its built-in calculations think you won't have any Ethereum left.