Along with the fine advise already given by those with more knowledge than I, here is a simple way I have found helpful in keeping my Handi rifle p.o.i. where it should be.
a) Cleaning- I only deep clean the bore with a solvent bath periodically as a minimum needed. Between at the range, hunting, etc. I run a dry (no chemical) bore snake through one time, every 7-10 shots. From the cold, fouled to the point of one dry bore snake pass bore, my point of impact is consistent for up to 10 shots before needing the dry pass again. This may not be doable if you have a noted heavy copper fouling caliber
b) OAL of cartridge- I just worked up a load this week. I noticed with the exact same powder charge and primer, bullet in my once fired neck sized cartridge, as I increased my OAL from spec to near maximum in my rifle and back. My POI groups went from vertical to horizontal. I settled for .016” off the lands. You may want to try seating your bullet .010” - .015” from the lands and see what happens?
If someone feels either of these examples would be of no benefit to the problem please do point it out along with "why". It seemed to work for me but I am not sure why from an engineering pov and always like to learn. Maybe more consistant pressure with the OAL closer to the lands in the long throated Handi?
If you try either and they help, lets us know.