The Colt Lawman Mark III was only available in .357 Magnum, not .22LR, .22 Magnum, or .38 Special, although like any .357 you can use .38 Special in them.
Only the Colt Trooper Mark III was available in .22LR, .22 Magnum, and a few special order .38 Specials made for police departments.
The .38 versions of the Mark III were the Metropolitan Police, the Official Police Mark III, and 496 Colt Officer's Model Match Mark III target revolvers.
First, I'd suggest buying a Brownell's bronze chamber cleaning brush and cleaning the chambers.
If the gun has been fired with much .38 Special ammo there may be a ring of fouling in the chambers that will prevent a Magnum from seating.
Use the brush by hand.
Second, get some factory loaded live ammo and try that.