We don't know if King David told Uriah to go home to be with his wife
euphemistically like you said because he wanted to kill Uriah someplace away
from the other soldiers, or because he wanted Uriah to be with his wife to
cover up her pregnancy by David. Since Uriah did not go home, we don't know
what David would have done next.
I know that Nathan the prophet said that David sinned by taking Uriah's wife
and having him killed, but I don't think it was considered adultery and
murder technically because Uriah was a Hittite, and they were not supposed
to marry Hittites, but kill them. Maybe David sinned morally, by having a
man killed for his wife, but not legally, because he was allowed to kill
the Hittite. Maybe that is why Psalm 51 says that David sinned in G-d's eyes
because G-d knew he did it for the wrong reason.