Easy, low fee way to move fro one paper wallet to another?
If you simply import your paper wallet's private key to another wallet, then you will not be protected against the backdoor that is being used to steal funds stored on paper wallets created with the code you mentioned. This is because importing the key will not move the BTC to a new address, it will simply allow the new wallet to spend the coins that are stored at the paper wallet address. Nothing would stop the thief from also being able to move those coins.
To protect yourself, you need to move the BTC to a new address, which was generated using secure software/hardware.
I would strongly recommend against using a web-wallet. Instead, I would recommend starting here: https://bitcoin.org/en/choose-your-wallet