What is the admonition to believers in this passage?
The admonition is that believers, having freedom, are to be led of the Spirit and live profitable lives that bring honor and glory to God by keeping old Adam in subjection. They are told to reckon themselves dead to old Adam and alive unto God through Jesus Christ, and not let old Adam reign in their mortal body.
Les teaches that the word "reckon" means to come to a mental conclusion. The believer has freedom now; it is not something commanded or demanded as already done, but is left with free will, a place of decision even as a believer. A Christian can choose to live a spirit-filled life or a carnal, fleshly life. The admonition is to be led of the Spirit, to live profitable lives, to bring honor and glory to the God of glory, by keeping old Adam in subjection. God has not taken away that free will from the believer. So the believer is to reckon himself dead indeed unto old Adam, but alive unto God through Jesus Christ our Lord. Likewise, "let not" implies choice: the believer is left with the choice of not letting old Adam continue to reign in his mortal body. Les explains that many unhappy believers are miserable because old Adam is still controlling them, and they are the ones left with the choice—though he doesn't have to.