If the cylinder is "going back and forth", this is known as end shake.
To measure how much is there, push the closed cylinder to the rear and hold it there while you gage the barrel-cylinder gap with an automotive feeler gage set. (Buy an auto gage set at most any discount or auto store cheap).
Then push the cylinder forward and hold it there while you gage the gap again.
Subtract one measurement from the other and that's how much end shake you have.
Colt specs are much tighter then S&W or others.
Factory specs are that the maximum end shake allowable before repair is needed is 0.003".
If it's over that the gun needs repair, and if you continue shooting it the gun will be seriously damaged and will cost even more to repair.
With the Colt revolvers the gun must be sent back to Colt. Repair requires the collar on the front of the cylinder be stretched with a special hydraulic device that "pinches" the collar and stretches it.
Then the collar is precision trimmed to re-set the correct end shake of near 0.001" or so.
Some notes:
Unlike S&W and most other brands, you CANNOT drop washers into a Colt cylinder. The design is quite different and putting washers into the cylinder will seriously damage the cylinder and possibly the frame.
With the cylinder pushed to the rear and held there the gap is the actual barrel-cylinder gap. The actual gap is NOT with the cylinder forward. A narrow gap with the cylinder forward is because of excess end shake.
A Colt should have a true gap of from 0.004" to 0.006".
While Colt has the gun they'll give it a total check out and internal cleaning.
Colt is back logged so repair will be slow.
If you just can't stand to wait, Frank Glenn in Arizona is a well known Colt expert who does factory level work.
He's faster then the factory, but Colt is still the recommended repair source.