So what are people's plans, anyway? I know that David is going to continue seeking apprentices, and it's very likely that he will confront Karza on every opportunity he gets. 

I think right now David's existence as the last Jedi is unknown by everyone except, now, Karza. Does anyone want him to do anything in particular, or for his existence to be revealed? Because one possible plan for that is this:

He's injured in his fight with Karza, and he flees and is forced to seek medical attention from a larger organization. They see his wounds and deduce that a lightsaber made them.