To develope a good drop shot from either side, change your grip to abut an western backhand. This means that when you put your hand like you are going to shake hands with someone, the racquet face should be horizontal. Hit the ball with a lot of underspin and a good arch, so the ball rises up about ten feet before the net and is dropping fast when crosses the net. In practice, you should get the ball to bounce at least 4-5 times in the service box regularly before trying the shot in a match.
Also, hit the drop shot down the line (over the high part of the net) , and only when the ball is short so you can come into the court and hit with more control and less arch. If you hit the drop shot down the middle, you make it to easy for the other player to get to the ball and too east to hit a response.
In reading your post, I would say you don't need a dropshot, you need a good slice backhand to change up the point. Your slice can change up the pace and location of the rally with a low margin for error.