Am using 110 grains of Goex 2f or Black Mag3 and 1 1/2 oz. shot. Lesser charges of powder didn't work at all. Range is 25 to 30 yards.
Less powder
More lead
Shoots far
Kills dead
More powder
Less lead
Close in
Wide spread
I believe the above rhyme, from an old Kit Ravenshear catalog(ue), has to do with the balance between moving the shot without blowing the pattern. It is counter-intuitive to adding more powder to reach farther. But first you have to have a load that patterns at least cylinder or it won't help. (Which is 35 to 40% of the pellets staying in a 30" circle at 40 yards). By the sounds of things you're not even at that density, yet.
In my 12 ga I get there with 82 gr FFg (3 drams) a 1/8" over powder card, a 1/2" cushion wad smeared with Bore Butter or rolled in beeswax/Crisco mix (Moose Snot when I use up these), 1-1/4 os shot, and an over shot card with a nick in the edge to let the air escape. No paper cup at all. When I want tighter groups, I scientifically underfill my measure to perhaps 75 grains and drop in another dozen or so pellets.
The shot cups I make lately are wax paper and are a tight fit at the muzzle. I get about 2-1/2 full wraps with the wood pattern I'm using to cut the pattern.
Gets me way up to an improved cylinder pattern, maybe 50% in a 30" circle at 40 yards. You'll never approach the 75 or 80%+ of a turkey tube in a modern gun. I'm wondering if the paper might not be working against you in the already choked tubes. I've never owned a choked m/l so I have no experience with them. It could be kicking the pattern to one side as it blows through the choke & shreads apart.
Here is a desperation method to try if you are willing. Use your dowel (or a large felt-tip marker/Hi-lighter tube) to shape strips of manila folder into 1/2" x 3" strips, bent in the middle to form a "U". When loading set two pre-bent strips at right angles "+" to form a cup and set this in (over the already tamped cushion wad) just deep enough to fill with shot, then tamp down with the over shot card. I think you'll have to add the shot at the muzzle to keep the petals from folding in and getting under the shot. Try a couple of these with maybe 95 gr of powder and 1-7/8 oz shot and see if it helps. This is an experiment, so you may need to alter the dimensions to get a good fit. The petals should be just long enough to hold all the pellets. The object is just to keep the pellets from touching the barrel and not to hold them any real distance in flight.
This is just something I thought up and has never been tested, so use your common sense if something doesn't seem right.
I'm finding grouse get more distance on me before I've got the gun set than they used to. I may try some myself now that deer season is over. This is my prime m/l time of the year. And it was 0