Trigger work can be intimidating. It is not that hard, if you have any mechanical ability at all. Print out the instructions and thoroughly study them, until you know them frontwards and backwards. Get the proper tools lined up, by that I mean have your slave pin made up and the punches, mallet, etc. I suspend the frame between two 2 X 4 blocks to punch out the pins that hold the trigger group in to the frame. I use nail punches to drive out the pins with, they are slightly concave so they will not slip off the pin while you drive it. After I get the pins well below the surface of the frame, I use a regular drift punch to push them out the rest of the way.
As to your zero drift, since you are assured it is not your mount or rings, I would look at either your scope or the latch engagement. Your scope could be drifting, they do funny things some times. It could also be your latch engagement. Make sure your latch engagement is the same each time you use it by making sure it is always dry from lubricates and you snap it shut with the same force each time. I have found that my shot groups have a tendency to string vertically with the Handi rifle, unless I do what I suggested.
Good Shooting and Good Luck