Let's say I have an exodus wallet and an electrum wallet. How would I go about transferring bitcoins from one to the other by use of exporting private keys without having to pay the transaction fees by sending to an address? I am the owner of both wallets and this is just to transfer to myself.
This is not a blockchain transaction, so fees are not applicable. You just transfer your keys from one wallet to another, which will allow you to use either of them. Your balance will still be available in both wallets, it won't double
As for how to do it you were already given detailed instructions. Good luck.