Yes as stated Paul includes himself by his use of the word "our"in verse 5 when he says, "But if our unrighteousness commends".
That comment is actually a continuation of the previous sentence. We know this by his use of the word "but" in the before mention clause. That connection to verse four is specifically in relation to the statement, "every man a liar" This he says in connection to who are living in unbelief in verse 3.
Anyway, Paul including himself as in unrighteousness, unbelief. therefore in part a liar. Living a lie because having sinned and falling short of the Glory of God. That is why he admits in verse 7 in the first person to lie. But he states this in the continuation of his argument from the flesh, the man.
In doing this, he also states that the truth of GOD; the oracles to which they were intrusted still got promoted. So from the flesh (the man) he says why am I still judge as a sinner? This is stated in relation to verse six. God judging the World. This is said in relation to verse four, "That You should be justified in Your words, and will overcome in Your being judged." Which is the whole point of Oracles in the first place.