Did you buy The Colt Revolvers: A Shop Manual, volume ONE or TWO, by Jerry Kuhnhausen?
You need Volume TWO which is for the later Colt revolvers like the Mark III.
Getting off-center primer hits is never good.
What this means is, on the rear of the cylinder you're getting off-center hits.
On the front end, the bullet is hitting the forcing cone off-center.
This can badly damage the rear of the barrel.
Plus, the gun is likely firing in an UNLOCKED condition.
NOT good.
There are any number of reasons why you're getting off-center pin hits.
First, does this happen when firing double action, or does it happen in single action too?
This could be a matter of the cylinder not rotating far enough and not locking up, OR it could be a problem with the cylinder rotating TOO far, and not locking up.
There's any number of reasons why this is happening, from a defective hand, to a bad ejector, to a bad locking bolt, to a bad cylinder to..........
Without actually seeing the gun, there's no way to diagnose it.
One "watch-out". These Colt's CANNOT be repaired by altering parts.
The parts are powdered cast steel with a very thin glass-hard coating.
ANY attempts to polish, cut, reshape, or bend a part will absolutely RUIN the part.
Any cutting or polishing quickly breaks through the ultra-thin coating, exposing soft metal.
Attempt to bend and the part just snaps in half.
These guns were specifically designed to be repaired by putting in NEW parts.
Old parts CANNOT be "fixed".
The way this is done is by pulling parts out of a bin until a "fit" is found, which restores the gun to the correct condition.
The problem is, local gunsmiths HAVE no bin full of parts.
They order a new part, but often run into the problem of it not fitting properly.
The temptation is to "make" the part fit by altering something.
This ruins the part, and although it seems to work OK, it's really not, and often winds up doing damage to the gun.
If you insist on trying to fix this yourself, here's the rules:
NEVER replace or alter ANYTHING unless, and until you're 110% CERTAIN exactly what the problem is.
Where big trouble starts is in guessing what's wrong, then fixing something that ain't broke.
Next, be aware that gun parts suppliers cannot supply parts for these Colt's by size. There ARE NO "over-sized" parts or sizes of parts.
Again, it's intended that the gunsmith have a bin full of identical parts, from which manufacturing variations will allow him to get a proper fit.
This is why I recommend sending these Colt's in for repair.
Colt and Pittsburgh Handgun Headquarters HAVE bins full of parts.