"I'm engaged on a new project" - means you are doing work on the project. 'Working on' or 'working in', generally can be used interchangeably, but "I have engaged on a new project", somehow doesn't sound correct. "I have engaged in a new project", sounds like you started working in/on a project.

As Wardaa said, "I've engaged in a new project" is the better of the two options, but it isn't very natural. "I've become engaged in a new project" would be better if you want to use 'engaged'. "I've become involved in a new project" would be better still.

I believe the first sentence is correct: "I've engaged in a new project." Here is why: [engage in something] to take part in a particular activity, especially one that involves competing with other people or talking to them. Here is an other option: " I've taken part in a project." I hope this was helpful :)