Was king David speaking of himself in Psalm 110, verse 1?

No. Please consider:

“A Psalm of David. The LORD says to my Lord: “Sit at My right hand Until I make Your enemies a footstool for Your feet.” (Psalm 110:1, NASB)

Explanation:

Firstly,  some Jews might argue that the word “seb” should be translated as “wait” instead of “sit”.

What they don’t tell you is that this same word is translated as wait in 2 Kings 2:4 but means more like sit in Jeremiah 36:15.

Perhaps it’s both “sit and wait”. Best is,  regardless the meaning is the same.

So,  Sit or Wait,  the meaning is the same.

David is speaking of two other distinct Persons when he says, “YHWH to my Lord”  (in Psalm 110:1) and certainly not himself.

The LORD = YHWH = Father

my  = king David

Lord = Lord Jesus Christ

(as only revealed obviously in New Testament Scripture in Matthew 22:45 or Luke 20:44)

Peace to you