Emotional connection - and expressing it.
I completely follow what you are trying to get across here, Old Hack, and for me at least, it's proving to be one of the most difficult aspects to get to grips with.
I can feel what the characters are feeling and fearing (or think I can - there's another issue) and think what they are thinking, but to put it into words that flow with clarity and sincerity - oh, how lucky those folks are who can do that with apparent ease.
We press on.
Worrying about voice, without the pursuit of clarity, flow, and the communication of emotion seems a rather futile exercise.
I find that if I get close enough to the thing I'm writing about, then the words just happen. It all flows, it is full of action and detail and texture. It is hard to get there: but once I do reach that point, the writing works (for me, at least!). And I'm not alone: other writers have described that state of writing in flow and have recognised the quality of the work they produce when they achieve full flow.
It's reasonable to assume that the creative flow-state is very close to the hypnotic state, and also to the meditative state. Now, I'm very easy to hypnotise and have learned over the years to get myself into something approaching that deeply relaxed state when I'm writing. When I manage it, I write my best stuff. I've also done quite a lot of research into writers' block, and have read several compelling pieces of research which say that block usually comes when writers are being overly self-critical.
(Stay with me here. I will get to a point soon.)
If block is caused by overly harsh self-criticism, then it can be cleared by practising self-compassion.
If the flow-state is a meditative, hypnotic state then it can be achieved by meditation and mindfulness.
There are lots of guided meditations online which focus on self-compassion.
I have helped several writers through the most horrendous blocks simply by getting them to commit to using those guided meditations every day. Five minutes, ten, twenty: find some that you like, give yourself a quiet hour without interruptions, spend time listening to the guided meditation and then write. It works.