Was King David baptized

Answer 1

In the Christian sense? No. David was Jewish; and moreover, he lived long before Christianity.
The earliest Christian baptisms, nine centuries after David, were (always, or at least very often) by immersion of the entire body (footnote 1, below).

Baptism by immersion is physically similar to Jewish tevilah, which is an act of immersion in water for ritual purification. King David, like every other Israelite, used this ritual purification on any occasion that Jewish law called for it.

Footnote 1:

Ferguson, Everett (1996), "The church of Christ: a biblical ecclesiology for today", p. 201: "The baptism commanded by Jesus... is an immersion in water. The topic formerly was debated, but these days there is general scholarly agreement."

Answer 2

The practice of baptism came centuries later, but he was anointed to be king, with drops of oil.