That's good.
David was not a priest at all. During his reign, he had several men serving as priest (Ahimelech, Zadok, Abiathar). David loved the Lord and sacrificed much, but did not cross the line into the role restricted to Aaronic descendants.
During the moving of the ark to Jerusalem, David made sacrifices and wore an ephod, but he is
never referred to as a priest.
Some translations say David's sons were priests in 2 Samuel 8:18, but there is a footnote on that verse with an alternate wording, "chief officials" (no mention of "royal priesthood" here). As David was not descended from Levi, his sons
could not be priests, so the alternate wording must be correct. Uzziah, descended from David, was king as recorded in 2 Chronicles 26. He entered the temple and burned incense... and contracted leprosy for violating sacred space as he was
not a priest.
Jesus, descended from David, is
the High Priest, in the order of Melchizedek, and believers in Him are the "royal priesthood", identified as such in 1 Peter 2:9.