Hey Erica,
After looking at the new Program and understanding what you are trying to do, I don't think that is the right Program for your initiative. It appears that the new Program is only for quick sends, you cannot amend the flow steps to account for the change in time zone. You simply pick a date and a time and off the email goes.
That being said, I would continue to use your Program for complex email sends so that you can amend the flow steps.
Also, the best way I have seen to handle time zones is to create a segmentation. You will be able to create up to 22 segments I believe, so not quite the whole world, but maybe you can combine a couple where you company doesn't really conduct business. The process to set this up can take a while depending on how many time zones you do business in, but after you complete it you won't have to touch it again. Then, when creating your campaigns, you can just refer to the segment and it will include anyone in that particular time zone. No refreshing needed.