I had this problem a number of years ago. In my case it was completely resolved by taking several steps. First, I dramatically cut back on hitting range balls. I now spend almost all of my time practicing my short game when I practice. Next, I changed my shafts to a vibration reducing shaft. At that time I did a lot of research and went to titanium shafts in my irons and graphite in my woods. I went to graphite as the quality improved and when I needed to change from stiff to regular flex. I would have stayed with titanium, but they were difficult to come by and had once again gotten very expensive. I did continue to play a steel shaft in my 54 & 58 wedges and still do. But, I almost never put a full swing on these clubs. Today I play graphite. I also went to OS grips. This change was in steps as I first tried midsize, but when I needed to replace my grips, I experimented with OS, I took the step to OS. Then I worked to stop taking large divots with my irons. It was early summer and it had been dry and with our clay soil in KY the ground was like concrete. This actually helped my game as I became more consistent overall with my irons as I learned to shallow out the bottom of my swing arch.
My experience was my sore hands and fingers was all about the shock of impact and the force of taken large divots. In my case it was not a gradual thing stretched out over a season. I was fine and then I noticed some soreness, thought it would go away and so I just cut back a bit on my playing. Not the case, it just got worse and worse in a couple of months until I took a complete rest for about 6-7 weeks and then went back with new shafts in my irons and a couple of fairway woods, working on a new swing to take no divot or at least a shallow divot, and replaced the standard size grips that put more pressure on my fingers than the dispersion provided by a larger shock absorbent style of grips. Think of the pressure of a Stiletto heel on a woman's foot. The force of her weight is the same in a standard shoe as in a stiletto heel, but the distribution to a small surface multiplies the force of the stiletto heel.
About 5 years ago I experimented with some steel shafts. It was a used set of some nice irons I traded for a putter I had. I inserted Pro Soft inserts, etc. I was thinking maybe it was OK. I had played for at least 5-7 years pain free. I did this because I never played as well with the alternative shafts as I had played with steel. Not a huge difference, but it was noticeable to me. I think it was the 3rd round I played with these irons I had my best round in several years, 2 under par. I was thinking great. However, it was two rounds later I noticed the pain in my hands and fingers.
Btw, I play the grips Sneakylong mentioned. Sneaky and I don't generally agree on many things, but I totally agree with Lamkin REL Ace grips. That is what I presently play in OS. I bought an extra back up set on sale a couple of months ago.