Recommend you get a real gun and not a cartridge conversion, here's why:
The frames are designed for MUCH lower Black Powder loads. These frames won't stand up to the much higher pressures that come with smokeless rounds, even the lower pressure smokeless "Cowboy loads" can and will beat up a frame.
Just how reliable is a Cartridge Conversion? Seeing as how they often require custom fitting by gunsmiths, do you want to trust one with your life when it came down to it? I wouldn't trust them except for clanging steel at a Cowboy Match, or punching paper.
There's also the matter of using Cowboy Ammo for self Defense. Wouldn't use it. VERY poor choice of ammo. Low velocity Lead Round Nose Bullets may make a hole, but they tend to be very poor stoppers. There are SO many better bullets out there than LRN bullets, I suggest you look into those.
Also, CZ-22 made a good point, the DA / SA revolver in .357 is a solid can't go wrong choice if you are intent on the revolver route. Very proven design / round combo.
Take CZ-22's Single Action Army advice with a load of salt though. Why limit yourself to an antiquated design that doesn't allow easy reloading, and with some designs like the Colt, Uberti, USFAs that force you to carry an empty chamber because of the firing pin mounted hammer? SAA revolvers are also NOT easy to conceal. The Single Action Army's time has come and gone. While still a nice pistol, it's a poor choice for a Conceal and Carry piece, IMHO.