One thing that you have to do is to remove the tumbler and polish the full cock notch. You will need a very fine stone with sharp edges so you can get into the full cock notch. Be very careful not to change the angle of the notch. Just polish it so it is smooth. You will also need to polish the trigger seer and you have to be very careful here, too so you don't change the angle of the seer or round it over. This is tedious and requires very careful work. While you are doing this, check to be sure that the full cock notch was not cut too deeply. If you have a deep notch, it can require more pull on the trigger to disengage it and result in a heavier trigger pull. Do not try to file away the front of the notch to make it shallower. If you do this, you will cause the trigger to catch in the half cock notch. If the full cock notch is too deep, you have to fill in behind it to make it shallower. I have done this using J.B. Weld to bond a tiny piece of brass shimstock in the back of the notch. It is pretty hard to get such a tiny piece of brass in place and hold it there. Plus, you have to properly prepare the surfaces to get the J.B. Weld to hold.
Once you have the lock taken care of the lock, you need to look at the trigger pin. If it is located too far back of the point where it engages the trigger bar, this can result in a heavy trigger pull. You may have to relocate the trigger pin to a position closer to the trigger bar. You just need to move it closer but not too close or it won't have enough movement left to move the trigger bar far enough to disengage the trigger seer. The closer the pivot point is, the lighter the trigger pull but the longer it becomes. It is a fiddly adjustment.
If the tumbler has a fly in it, you may be able to put in a set trigger. Doing that, would solve your problem with the trigger. It may require some inletting and adjusting but if you have a fly in your tumbler, it can most likely be done.