The more regions you control, the more family members you get, that much is certain. I'm pretty sure protectorate cities you have under your control counts towards the total also. I had an experimental campaign back in January, where I had lots of resources but only 3-4 settlements. My family tree was limited and ageing. I then got Seleucid and the Britons to become my protectorate, and possibly another faction I can't remember which one. Suddenly I had over 35 regions under my control, and new adoptions and family members were being born nearly every turn until I had well over 20 family members in total. When I gave the Seleucids some money to stop being my ally, I had less regions under my control now and no new family members. I suppose the game needs to limit things or you could in theory end up with 200+ family members by 14 AD or whenever the game ends. Not having any more births just because your empire is small is a bit crap though. Maybe there should be more births, but far greater chances of them dying young.

And to answer the original question: move on and expand, even without generals. Your men will take charge of things until a new general pops out of some filthy Roman lady's womb.