Hi Adam,
The biggest problem I see is that you aren't getting your shoulders extended at lockout. This is why you are having trouble straightening out. The barbell needs to go back directly over your shoulders and your torso (not your neck) needs to go forward to make this happen.
Try taking your press grip on the chinup bar and hanging straight down. See how much more open your shoulder is? This is what we want in the press.
Other than that, you're bending your knees quite a bit. If you're going to do the dynamic hip motion in the press, don't hold at lockout and breathe, but bring it down to the bottom position. Big breath, tight abs/knees, then Hips--press.