If I move my data to Altium 365, will I have to design boards in the cloud and be connected to the internet to get any work done?

No, you will not design boards in the cloud. Altium 365 adds data management and collaboration capabilities to your design workflow, but it does not disrupt or change how you design PCBs in Altium CAD software (Altium Designer and CircuitMaker). You will continue to do schematic capture and board layout in Altium Designer or other Altium PCB design products. When you open a project from the cloud in Altium Designer, a local copy is created on your machine. You can work with the project offline for as long as you need and then save changes to the cloud server. You can work offline (see more details on Working Offline) for periods of time when your internet connection is not stable, for example. You do need to connect to the internet to share your work with others and collaborate.