of David = by David. The words alleged to be Chaldaisms in verses: Psa_139:3, Psa_139:4, Psa_139:8, Psa_3:20, are found in the earlier books such as Lev. 1 and 2 Samuel. There is no internal evidence of non-Davidic authorship.
LORD. Hebrew. Jehovah.App-4.
searched = search out as for treasures or secrets.