Sento and idaho are right on, but I'll add some related thoughts.
Volume of punching and training is a consideration. This is probably leveled out after a year of training the new stance but it bears mention nonetheless. The other thing is that your muscles and structure have probably not adapted yet and you may not be training them adequately on that side. If that is the case then an overcompensation is occurring, which also goes to Sento's point about a form/technique issue, probably subconsciously.
An alternative scenario might be that your left side and tissues are tighter and more knotted than your right side. When tissue gets knotted up and/or tight it also exposes the connective tissue to more stress than it otherwise would see. This might shift it from being a load your body can handle to a load that is causing pain due to overuse or inability to recover.
Feeling this in your shoulder is an indication that you are possibly using to much arm and not enough hip/leg when throwing. In addition it may be that your back musculature is too weak to properly decelerate the load from your hands and that your shoulder and elbow are compensating for this by tightening up (again, what I said above). This is not really uncommon with a non-dominant hand. Typically people's dominant sides have more muscle and/or more efficient recruitment patterns.
This is also physically explainable: with a lead hand punch you are generating less power and velocity than you are with your rear power hand. The hips and legs by definition cannot load as much force into your hand due to less distance, and less hip chambering/whip over less distance. The rear power hand has years of practice in throwing the hips and legs ALL the way behind the punch and over a greater travel distance. This also means your back musculature more practice in decelerating the hand properly.
Now switch them around: your weak side is suddenly thrown a much more challenging state of affairs by switching stances. The muscles that properly decelerate punches are not strong enough or experienced to do the job properly.
TLDR: work your back--all of it--on the left side and get it stronger combined with more conscious work on hip/leg and technique.
I'll defer to the technicians here on punching.