No gain in pain
Sounds like part of the problem is the S&W 625 is too large to fit your hand. lets start there. Place the grip of the gun in the web of your hand between the thumb and index finger, high on the back strap. Align the barrel with your forearm. From this position you need to be able to reach the outside of the trigger with the crease of the first joint of your index finger - sometimes called the "power crease". If you can not reach the power crease from this position the gun does not fit. You are forced to rotate your hand so the back strap is behind the thumb knuckle in order to reach the trigger. The thumb knuckle will not absorb much recoil without getting sore.
If you are using wraparound grips, try a pair that are flush with the frame and see if that allows you to reach the trigger correctly. If this still won't work, you will have no choice but to rotate your hand until you can reach the trigger.
Grip material can not be slick. I sanded the pretty shiny finish off my wood grips. Rubber works quite well in this regard.
When you shoot, grip the gun very firmly with the shooting hand and curl the thumb down. Place the support hand in front of the shooting hand, knuckles over knuckles with the index finger firmly against the trigger guard. The support thumb curls down over the shooting thumb, thumbnail over thumbnail. While maintaining a firm grip with the shooting hand, lock the gun in with isometric tension by pushing forward with the shooting hand and pulling back with the support hand, firmly. You can do that with either a Weaver or isosceles stance. Now press the trigger until the gun fires.
Follow through. Don't relax the grip the instant the the gun fires. If you do the gun will squirm and pull away from the support hand forcing you to re-grip after every shot. Maintain the grip pressure and isometric tension through the recoil. This will greatly reduce muzzle rise and pull the gun back on target near the point of aim in an eye blink.
Start slow. Begin with some dry fire. Use the .38 until you get the hang of it. You may or may not be able to do well with an N-frame size gun. If not, don't sweat it. Everyone need a gun that fits their hand in a caliber they can control. What someone else can do, or claim to do, is irrelevant.
Let me know how it works out for you.