Until you have access to your phone number, there is nothing that can be done with Signal.
Once you get a new phone you can install Signal and register. This will stop all messaging and calling on your old phone. Your past message history will remain on your old phone. All new messaging and calling will be handled by this new phone. A new install of Signal also generates a new safety number for each conversation thread with your contacts. Your contacts may need to, depending on their settings, accept the safety number change to be able to see any new messages from you and send new messages to you.