Check end shake with the gun un-cocked and the action at rest.
Checking with the trigger back tells you nothing, and will actually give you a false reading.
Unfortunately, correcting end shake in the Colt revolvers is much more difficult than in S&W and most other brands.
On those, you can stretch the crane shaft or use washers to correct excess end shake.
All these brands establish end shake at the REAR of the cylinder by the fit of the cylinder shaft against the inside-rear of the cylinder.
The Colt establishes cylinder end shake at the FRONT, by the fit of the cylinder collar against the crane shoulder on older revolvers, and by the fit of the collar against the frame in newer guns like your Trooper Mark III.
In the Colt's, end shake correction is a factory ONLY job, or one of the VERY few service centers who have the highly specialized equipment to do the job.
In the Colt, the cylinder collar is stretched to correct end shake.
This is done by putting a hardened steel support inside the cylinder, and using a special custom-made hydraulic "pincher" tool to squeeze a ring into the collar.
This is very "ticklish" work, can only be done ONE time, and can only be used to correct a limited amount of end shake. There's a chance the collar can crack.
If the collar can't be stretched for any reason, the cylinder has to be replaced.
Since this is NOT something any local gunsmith is going to have, you should trust ONLY the Colt factory, Pittsburgh Handgun Headquarters, or Cylinder & Slide to do it.
In general info on the Colt Trooper Mark III, this is an extremely tough, strong, high quality revolver. Quality-wise, the finish, fit, and accuracy is better than most anything made today.
The only "watch-out" is to use snap caps if you dry fire.
Some firing pins may break, and replacement is another factory ONLY job, since replacement takes a special press with support jigs to protect the frame from being damaged.
The Trooper Mark III was introduced in 1969 as a replacement for the older Colt revolvers like the original Trooper and Official Police models.
These older guns had to be hand fitted by having each part stoned and filed to a fit.
This extensive hand labor cost a lot of money, so the Mark III series was introduced as a replacement for them.
The Mark III had parts that were made from an early version of MIM (Metal Injection Molding).
In this process, parts were made from powdered steel injected into a mold and heated until the metal fused.
These guns were assembled by test fitting parts from a bin until a fit was achieved.
The Trooper Mark III was replaced by the Mark V in the early 1980's, which was simply the Mark III with a short action for a better trigger pull, and the Mark V was replaced in the mid-80's by the King Cobra, which was simply the Mark V made in stainless steel with a new lugged barrel.
