How can I create separate rooms and allow users to freely move between them?

Jitsi Meet (an open-source Zoom alternative, https://meet.jit.si/) lets you connect to a meeting with just a link. It should allow you to create as many meetings as you want so that the judges can just click on the next project's meeting link. You could give each team their own room, and from my testing you should be able to be in as many rooms/meetings as you want at the same time.

It's a little confusing, but co-hosts in Zoom can move between breakout rooms. They just need to enter the room the host assigned them to first, then they can switch rooms by opening up the breakout room window like so and clicking on "Join":

(Screenshot from Tufts)

This works best when you have only a few people switching rooms as it's a bit tricky to expect users to do, even with instructions. It's also annoying to set up a lot of co-hosts and potentially insecure.

What I have seen done is having multiple Zoom rooms open, and the participants are given a list of Zoom links that they can join when they want to. (Usually there is a main session that everyone joins to start out the event.) This setup comes with different challenges (the host needs to pay attention to let people in usually, you may not have Pro licenses for each host, etc.) but is easier for people to access.