I would like to use my video signal for projection on a screen, while also using it to send a live stream. Is that possible?

Here are a few ideas on what you could do (depending on your camera setup, type and plug types):

Use a simple cable splitter to split your video signal between live-encoder and projector (such as a HDMI splitter)
Use a video switcher with multiple outputs
Use multiple outputs of your camera, if your camera supports it
Use your hardware encoder’s output to run a cable to your projector
Use your graphic card’s output on your computer if you are using a software encoder on a PC or Mac.