Would the professional thing to do be confronting my boss, explaining
the other projects I'm on (he currently only has a vague idea), then
asking which should take priority (the new project or the existing
Other than the term "confronting", this is exactly the right approach.
In many roles, it's completely normal to be working on multiple projects at the same time. But it's important to understand what has been promised, what are the schedules, and what are the relative priorities. And it's important that your boss have more than a vague idea about what you are doing.
Work together with your boss to make sure you understand how your limited time should be allocated. That way, if deadlines get missed, the most important work will be done at the expense of the less important work.
You need to do this periodically. If you have a weekly one-on-one meeting with your boss, perhaps this should be one of the topics. It certainly should be discussed every time a new task is added to your plate.
If you don't already do so, you should send your boss a weekly Status Report. In it, you should show what projects you are working on, where your time is currently being spent, what your current estimates for completion of those projects are, any upcoming projects, any upcoming planned time off, and anything else your boss needs to know.