i used to have the same problem, but i figured out a way to lift the dumbells by lifting my thighs to get the weight up.
however you did say that you only do the press from the elbows no lower than your shoulder blades, this is the stronger portion of the lift so when you use weights that you cannot lift in the lower, or weaker portion you might not be able to get the first rep.
The best way to solve this is just to use lighter weights in the lower portion of the lift, remember, it is form that is important, not the amount of weight you can lift.