Staking when moving to cold wallet

Hi this has sort of been discussed but I’m now 100% used to cold storage. I’m using Tangem for some of my stuff. Now I have DOVU staked from my Hashpack and want to move it to the Tangem. Do I loose any time rewards when moving? Also do I just register the public Tangem id on marketplace. The idea of having any cold address kinda public bothers me. Howto on moving in stepwise simple way thanks