I would think that the only way that injectors could leak fuel is if there is something wrong with them, but then you'd likely notice other problems. But I'm kinda on the fringe of my knowledge there.
If your batteries aren't tied together in any way, then you won't have a battery selector switch. Obviously I'm not there to help, but from what you describe it sounds electrical to me. Sounds like one battery is not so good (maybe a bad cell) - when you use the trim (draws a lot of power), it starts to drain the battery. When you let it set for a bit, it recovers enough to crank faster. Just because they're new doesn't mean their good. Unless you load test them properly or remove the cables and let it sit for 24 hours before checking with a meter, other ways aren't very accurate. AND, if you do have a battery switch and it's set to "both", you're getting an average V reading of the two batteries.
Corrosion can also work it's way up underneath the jacket. How old are the cables?
Switch batteries and see if the problem follows the battery or the engine. This is just basic diagnostics 101 to eliminate variables... starting with the easy things first.