Misfiring in double action is a classic symptom of a mainspring that's been bent to lighten the trigger pull and bent too much.
While bending the spring IS a valid pistolsmith technique on the old type Colt action, the reason for problems is that the job is done blind, without using a trigger gage.
The bend is done by installing a small diameter rod between the rear of the two "legs" of the spring and cocking the hammer to put a slight downward bend in the upper leg of the spring.
If that's not enough lightening, a slightly larger pin is used.
People get in trouble because they start out with too large a pin and have no idea what the weight of the trigger is because they didn't use a gage.
result is too light spring tension and misfires in DA, if not in SA too.
You can sometimes ID a "tuned" spring by a slight downward bend in the upper leg of the spring.
One aid in diagnosing your 357 problem is to use a trigger gage to weigh the double action and single action pull.
The "I" frame 357 and Python are the same action so measurements are valid for both.
Double action....Maximum...12 pounds.
No minimum.
Single action...Maximum...4.5 pounds.
Single action...Minimum...2.5 pounds.
Note that single action and double action are both affected by the mainspring. Lighten one and you lighten the other.
Inspect the firing pin for chips or breaks. Happily the 357/Python firing pin is easily changed by removing the rear sight and sliding the cover plate up.
Check the pin and spring tension for fouled parts. If necessary disassemble and clean the pin and spring and the seat in the frame.
NOTE: On most all Colt schematics for the 357 and Python the firing pin is shown BACKWARD.
Years ago a technical artist mistakenly drew it backward and everyone just reuses the same view.
To be clear, the cone shaped firing pin spring small end fits over the firing pin with the wide open end facing the muzzle.
Best advice is to try some different ammo. If it still misfires, replace the mainspring with a new spring.
Most new springs sold today are actually non-Colt replicas but are factory equivalent.
If none of this cures the issue, I strongly recommend you NOT take it to any local gunsmith.
Very few these days are qualified to work on the old Colt action and often you get a gun back even more botched up.
If repairs are needed, send it in to Colt, or to Frank Glenn Gunsmithing in Arizona.
Colt is the go-to preference, but Glenn is a famous Master who does Colt factory level work.