I too was in a similar boat as you. I had plenty of training in the neurosciences beforehand. I loved my psych rotation and thought I could be a very effective psychiatrist, especially if I went to a place that emphasized psychoanalysis (Columbia to name just one off the top of my head). However, after seeing more and more patients in the outpt setting, I realized it was the pts who came from well-to-do homes, etc., and were educated, but had issues, wanted to talk about them, but didn't do what it took to get over them/resolve them/adapt frustrated me the most. I realized that over time this would begin to really bug me, and that I wouldn't be able to leave my work at home.
Although this is a huge simplification/generalization and I apologize for not stating this better, most of the neurology patients you see, while they might have some of the "self-affected" conditions you see in psych - depression to name one - are usually suffering from primarily physiological conditions that they can't quite readily influence. You can't exactly will yourself to have a stroke or have myasthenia gravis or Parkinsons or AD. These conditions I could go home at night and not worry about overnight. And, I could still interface with lots of the parts that overlapped with psych. Plus, you have to ask yourself what would get you out of bed in the middle of the night? For me, answering a stroke page, myasthenic crisis, acute GBS, etc., were at the top of the list.
As far as the dual-residency, I think it really depends on your area of interest, desired clinical or academic practice, and frankly how busy you want to be. You could do a cognitive neuro fellowship and end up seeing many of the same types of pts as the psych docs - AD, PD, etc., but from a different perspective.
Either way, I don't think you can go wrong. You just have to prioritize what your goals are and what'll make you [begrudgingly happy] get out of bed to answer a pager. With either, I think you can dabble in the other, though with different caveats. For my interests, I'd rather be on the neuro side and then cross over for certain things.
I chose neurology and at this point - just graduated med school - couldn't be happier.