Picard is canonically an Admiral by 2381, based on the fact that he was wearing his Admiral's uniform in the picture of him holding baby Thad Riker. Which, yes, is still a year away from when Lower Decks is currently set.i don’t think Picard is on the verity yet. According to the serie’s background material (which is not canon but makes a lot of sense) he will still be captain of the Enterprise until the mid 2380s.
Thinking of it -and I just checked- the Verity isn’t canon at all for the moment. All we know is that by 2385 Picard was an admiral and had left the enterprise and then resigned.