Always a challenge to get dialed in on a new bike (IMO) Sounds like you are doing quite a few things at once. Things I would consider 1. have someone look at you riding, maybe do a video so you can watch it.
2. Look at the handlebar to saddle height, Personally I ride with my handlebars level with my saddle height but also have friends who ride with their handlebars slightly higher than their saddle. When someone is just getting back into riding a bike we'll often suggest that position as they work on strengthening their core. Plus you aren't racing, the goal is to be comfortable and enjoy riding.
3. If you are too stretched out that is going to bug your hands as you put more weight on the arms and hands than on your saddle. While on the trainer, try closing your eyes and moving your arms/hands to where you feel most comfortable - if you find yourself just touching your handlebars with your fingers consider a shorter stem, they aren't that expensive and easy to switch.
4. Saddle - if you are sliding forward on your saddle you'll be putting pressure on your arms and hands. Prior to changing out saddles (although I wouldn't rule that out) try tilting the nose up or at least level. I ride a leather saddle and mine is neutral (level) but my DH's saddle has an upward tilt which keeps his sit bones resting comfortably on the saddle. Yes it sounds counter intuitive but give it a shot.
5. Saddle height, I rode my Surly ECR yesterday after spending a month on my touring bike and realized my saddle is a smidge higher on the ECR but I didn't change it. After 14 miles I noticed my right hand was have some numbness and I think it happened because my saddle was just a little to high, shifting my weight forward just enough to bug my hand.
Just a few thoughts, sounds like you are on the right track.