I just suggested what I thought would be easiest. You can import your wallet into the node software if you want.

and yes you can send them to a new wallet if you want, but it isn't any better or worse. The only thing you're doing by sending the coins somewhere new is defending against someone taking the valid signed transaction from one chain and broadcasting it on the other. I guess if you send to a new wallet there's less footguns to worry about?

To get into the weeds, you could also send the same UTXO to the same address, but just with different fee rates on each chain. As long as the newly created UTXO is different in some way, but I don't suggest that because of the higher chance of footguns if you do it that way.