I never heard it taught that Christ became a sinner. I agree that is a false teaching.The sins of God's elect were not transferred to Christ. Christ was not made a sinner. The sins of God's elect were imputed(credited, charged) to Christ and He paid their sin debt; but it was by imputation only, and not the false teaching by some that Christ actually became a sinner.
Likewise, Christ's perfect righteousness was imputed(credited, charged) to God's elect, but was it was by imputation only, and God's elect do not become righteous in themselves. The righteousness imputed to God's elect currently resides in Heaven in Christ Jesus, seated on the right hand of God the Father.
Whatever way Christians believe Christ bore our sins is the same way we bear His righteousness.
I believe Jesus told us this way in John 17. The early Christians viewed this as solidarity (our translations use "unity"). Jesus bore our sins, was made sin on our behalf, came in the likeness of sinful flesh, was made like the brethern, and referred to Himself as the Son of Man. This is solidarity or unity. We will be made new creations in Him, we are goreknown in Him, we are predestined to be conformed into His image, we will be perfected in untiy with Him. This is solidarity or unity.