Actually, the second coin was supposed to be a "Tiberius 579-582", that is, Tiberius Constantine

but I agree that Maurice Tiberius is more likely.
The (fragmentary) legend I'm seeing on obverse doesn't seem to fit either, or any emperor at all, but I might well be reading it incorrectly (or it might have been blundered in the first place); I agree that I probably should take the coin out of the holder. I don't like coins entombed in plastic anyway.
I agree that the reverse on the first coin is almost identical to the above example [on second thought, there are some minor differences - see below]. Obverse legend has to be set up differently - if that's a CAES above the portrait's head (which is possible though I'm seeing an O personally), the A is about where your coin's C is!
The more I look at my coin's obverse, the more I'm seeing a legend nearly identical to yours starting from CAES - but offset roughly 1-2 letters left. Looking at Wildwinds, this means that I have a RIC 111, which in any case fits the reverse (even) better.
So that's a Claudius. Which is probably cooler anyway, and in any case first century.