Perfect practice makes perfect
(wow, look at the font menu stuff for TFL now! I like it)
Dry fire practice is the best way to get good with both, or either hand. When you dry fire, study everything about how you're doing grip, trigger press, sight picture, etc. I found through thousands of dry fire reps what sort of grip and trigger manipulation seems to work for me with my chosen handgun.
As one example, I found that I have best trigger manipulation with my Glock 21 if the trigger face is outside of my first joint by about one trigger face width. So it's not quite the first joint and not the center of the pad. Also, I had to do a lot of work to find what sort of angle my fingertip makes for best effect. Too much curl sets up a sideways force on the trigger that makes the gun shudder from side to side when the trigger breaks. So my finger makes a sort of perpendicular contact across the front of the trigger. Another thing I found was that the human fingertip wants to move in an arc but the trigger is designed to move straight back. Therefore, a slight unnatural crunching move of my trigger finger to avoid curling works for me.
I hope your handgun has a short trigger travel. I tell people contemplating new handgun purchases to avoid double action autoloaders. That knocks out about 80% of the choices. Although it is possible to get good with a double action autoloader, it's a huge amount of effort and most of the people I've met who got to that point just went to a Glock or 1911. Some pistols like Kahrs have the same trigger pull for every shot but the travel is still pretty long. Actually, the first trigger pull on a Glock is different from the rest if you're shooting efficiently but it doesn't bother me or anyone else I know, unlike most other double action pistols.
Note that to get good, it helps tremendously to pick ONE handgun and use it for every handgun role you have.
Mental attitude is very important. It took me years of shooting to get to the mental focus where I try to make each shot perfect. I note that most people who throw a lotta lead downrange don't seem to mind inferior accuracy because they have lots more where that came from. When I took the attitude "THIS IS MY ONLY SHOT" then I got much better. Along these lines, huge numbers of dry fire repetition is not the best practice. If you do 25 perfect ones a day, then that's real progress.
Along these lines, my dry fire is in 50 "click" sets, but I may actually click for 200+ because I'm not counting the sloppy ones (where I can see the front sight move as the trigger breaks). I'm trying to reinforce the good points of my practice, not the mindless count to some set number for the day.
One other thing I do is when I get home I unload and dry fire ONE time. I view that as "this is the only shot I'll get so I have to be perfect because there is no second chance". This emphasis on perfection from the start seems to help me.
One last pointer: Try moving your trigger finger in a trigger pressing motion without moving any other fingers of that hand. It's not learned overnight. You can do this anywhere though you might appear strange to some.
I probably blew your mind with the details. Don't try to emulate me 100%. Find out what works for you.
Whew, how did I get so long winded??
Edmund
Are there any exercises or drills I can do to improve my weak hand hold? Any thoughts would be appreciated thanks 
