In reply to WB:
Hi WB,
Your dominant side tends to do a lot of fine motor control actions such as writing or feeding yourself with fork. As such muscle recruitment and movement patterns are much more ingrained that with the non-dominant side. In essence, doing stuff feels more natural to the dominant side.
This could lead to the perception of weakness as your technique is not as refined, the recruitment is not there when you want. Yes there will be some difference in strength but unless you are built like a fiddler crab or Nadal its not something you cant educate your body to overcome.
So do your offset pullups, make sure you strengthen your shoulders, work on your core. Always do the reps/sets with your weak side first.
Shoulders: google scapula pull ups, dips and pressups. These exercises will get your large back muscles engaged properly and permit a good clean pull up. Offsets put a huge strain on a small section of the body which gets neglected. Do 10 reps per set and do 3 sets. Add weight once you can complete this without failing in the last 3-4 reps of the final set.
Core: side plank, plank, back bridge. For the first two you want strict form and do this with straight arms - none of that rest on your elbow stuff. Work up to being able to hold each pose cleanly for 2 mins.
Pullups: Narrow, regular and wide offsets. Also if you have access to a set of rings, do typewriters on these and allow your wrist to rotate, it will save you the dreaded elbow tendonitis that comes from torquing your elbow hard on a statitc bar. For strength I would go with 6 reps per set and perfrom 4-5 sets with around 4-5min rest between. Do more and you work endurance, do less and you work power.
Fingers are often neglected as well, my advice is to deadhang but do so in offsets, i.e non-dominant hand hangs the crimp whilst the dominant holds a low hold with say 1-2 fingers. Deadhang protocols can be easily picked up off the beastmaker, metolius etc websites. Consider trying Eva Lopez maximum added weight, minimum edge width protocol, once you hit 40-50kg added weight start hanging one handed.
If after a few weeks of trying all of this you make improvements, this will be neuromuscular improvements. After about 6 weeks your actual muscles should have started adapting so you should really see some improvements on your former pre-trained self.
Good luck and remember if it was easy everyone would be doing it.