If you feel good about the scope and alll screws are tight then I suspect its because you quit swabbing between shots, the barrel has to be conditioned the same way each time to get a good group...With a flintlock, in hunting conditions I can get by without swabbing because I am loading a patched ball inside a greased patch. This helps clean and lubricate the barrel when I load...With a inline the fouling in the barrel builds up as you shoot, sabots are not lubricated, so with each shot the condition of the barrel changes... When I shoot in competition I always clean my flinters between shots, it makes a difference...
When I am sighting in an inline I have a spray bottle with rubbing alcohol that I spray two patches with, they should be damp, not wet as this will build up in the breech plug and contaminate the next shot...I run both patches down the barrel and use both sides this is followed by a dry patch and then a lightly oiled patch followed by another dry patch this ensures that the barrel is the same from shot to shot....
I have sighted in several Omegas for friends of mine, I have had good luck with the 250 T/Cs and 90-110 grains of Goex FF or Pyrodex R-S...I don't care for pellets as they won't let you fine tune your load to the rifle...I also don't care for Triple Seven because of the crud ring...In addition, some Omegas have tight bores, after the season you might want to try the Harvester Crush Ribs or the MMP-24 sabots...I usuallly buy my sabots in bulk so I know they are the same...Sometimes the bullet/sabot packets come with different thickness/length sabots and this can make a huge difference in accuracy...
As far as getting off a quick second shot with a muzzleloader, forget it, make the first shot count, carry a small bottle of alcohol and oil along with a few patches and clean again before you load.