The easiest way to commit yourself to leaving a job and to navigate explaining yourself to the company happens to also be the way you should always be leaving jobs: by quietly interviewing for other jobs.
Find a new role somewhere else that you like and accept it. Go back to your current team and inform them that you've enjoyed your time working with them, but you can't pass up the new opportunity.
The only problem I think you have here is that you haven't started interviewing yet. So, of course you feel aimless and uncertain about your next step.
Agreed, definitely, you should start interviewing immediately.
I appreciate that you want to stay friends with these people, and maybe you can, but then again maybe not. It is totally unprofessional and utterly silly for someone to get angry when one of their people quits, but nonetheless I have seen it happen. I would suggest that if they react that way, maybe they aren't such good people to have as friends after all.
As a new engineer without a significant work history, how does the poster interview and not have the hiring company contact his current company for a reference?
By asking them not to contact his current company. This happens all the time.
I really don't understand why there's any drama here at all. As a matter of course, you should have your next opportunity lined up before you inform your employer that you're thinking about leaving.
In fact, in this scenario he can be completely honest and say that that would not be a good idea because, well, he's the main engineer there and they would not be happy about him leaving. I know for me that'd be a very good sign (assuming I believe the guy is being honest).
Often times the prospective company will ask you outright if they can contact your current employer. Companies with a formal application process may prompt you in the documentation they ask you to fill out.
Regardless, there is no harm in saying no. It's the norm for folks that are exploring other opportunities. The only time it may be questionable is if you've already left (or been let go by) your previous employer.
Find a new role somewhere else that you like and accept it. Go back to your current team and inform them that you've enjoyed your time working with them, but you can't pass up the new opportunity.
The only problem I think you have here is that you haven't started interviewing yet. So, of course you feel aimless and uncertain about your next step.