Thank you for your reply. I'll check it out, but, unfortunately (for me), I think the Charles Beach you found is probably not either of the ones I'm looking for. The closest would be my Great Grandfather, Charles Nathan Beach, the silversmith, but he was married in 1874. Making an assumption that he was about 20 years old (I actually think he was a bit older because my Great Granmother was 28 when they married), that would make him 47 in 1901, long in the tooth to join the military, and extremely old at his retirement. He (your Charles) is also not my Grandfather, although I suppose he conceivably (so to speak) be a half brother to him.
Charles Isaac, , his father, the sailor, might not have been listed in any Connecticut Army units, as he was a sailor, even though serving with Soldiers. He also could have served with Federal forces. and not just State. I have been told, his portrait hung in the Statehouse - so this is something I also have to ask the Connecticut Historical Society about.
There appear to be more than a few Charles Beachs, so I'm afraid I'll just have to continue my search. Thanks, again, for replying. if I find out anything additional, I'll post it here.