If we are to be baptised "for the remission of sins," then where does that leave Jesus' sacrifice on the cross?

Why was Jesus baptised by John? Jesus had no sin.

I was taught that to be baptised by immersion is following Christ's example and an outward action/testimony of having accepted Jesus Christ's atonement and therefore leaving the old, sinful self/life behind (burying it) and being raised to "a newness of life," ie "being born again" to a new life in Christ. (Gives me chills just thinking about it!)

I was baptised as a baby in the Presbyterian church, but I was later baptised by immersion as an adult.