This sounds like a postural issue. Most people drop their hand because their elbow is in the wrong place. If your elbow is at your side, you hand should be near your chin. When you punch with the left hand, the right hand shouldn't have to move. Most people who flare their elbows drop their hands once the elbow is away from the torso. Stick something under your arm and hold it while you jab, but relax. Don't flex to hold it.
If the chin keeps coming up, that's usually due to hunching. If your back curls, you can't keep your chin tucked.
Power from any punch comes most efficiently from the movement of weight, not from how hard you move your arm. By default the arm and shoulders aren't very strong. The shoulder is actually quite delicate of a joint. That said, if your weight moves only FORWARD with your jab and your head does as well, you compromise distance. In other words, it's easier to hit you back.
To put it very simply, while you're jabbing, try performing like a half squat (with your left foot moving forward) but keep your head closer to your right foot. It will likely feel very weird at first, but when you hit something you'll understand why. And when you hit someone and they try to counter, you'll understand it even more.