What does Paul constantly tell believers about old Adam?

Paul's constant admonition to believers is not to yield to old Adam or give in to him, but to keep him under subjection. Paul applied this himself, keeping his body under control so it would not become an instrument of unrighteousness.

Les explains that Paul's letters repeatedly admonish the believer: do not yield your members as instruments of unrighteousness unto sin or again unto old Adam. The constant instruction is don't give in to old Adam; keep him under subjection. Paul had to do this himself, and he said in Corinthians that he kept his body under control lest it become an instrument of unrighteousness. Les clarifies this does not mean becoming a cleric or monk or torturing oneself, because the Christian life is a life of freedom, joy, responsibility, and fullness.

What is Paul's constant admonition about the old self or old Adam?

Paul consistently admonishes believers not to yield to old Adam, but to keep him under subjection. He urges them not to give their members as instruments of unrighteousness unto sin. Paul applied this himself, keeping his body under control so it would not become an instrument of unrighteousness.