Hello

I want to move my btc from my non segwit wallet to a new wallet that is supporting segwit.
Apart from sending them as a normal bitcoin transaction that would cost me some big money (my wallet does not support custom fees), if I am not wrong it is possible to move them using the private keys and save on the fee?
How does this work? Something I should be aware of? Is it safe to do so?

Thanks a lot!