Is foot washing a form of sin cleansing? I'm currently studying John 13:1-20.  I go to a church where they believe that foot washing is an ordinance similar to baptism and the Lord's supper.  I personally disagree and believe that as Jesus said, v.15 (ESV)

For I have given you an example, that you also should do just as I
have done to you.

I believe that Jesus is demonstrating that if the greatest (being Jesus) is willing to serve the lowliest of tasks, we too should be willing to serve one another in humility. So I understand that as acts of kindness and service (which can obviously take many forms).
Where I'm really struggling from an exegetical perspective is in v10 which says

Jesus said to him, “The one who has bathed does not need to wash, except for his feet, but is completely clean.

My assumption is that Jesus meant by you are already completely clean, is that by faith in Christ they have been cleansed of all sin, and have transferred from the kingdom of darkness to the kingdom of light.
So my question finally is, what connection does the symbol of washing only the feet have to do with the cleansing of the overall body.  If the cleansing of the body is the forgiveness of sin by grace through faith, then what does the washing of the feet symbolically represent, and how does that link back to forgiveness of sin? I hope that makes sense.  If you need any clarifications please let me know.
The story of Jesus washing the feet of Peter and the other disciples has two related meanings.
When Jesus washes Peter's feet, Peter objects, since he believes Jesus should not be doing such a lowly task for him. Jesus response tells Peter that this washing is an integral part of Jesus and Peter's relationship and so Peter enthusiastically accepts not just the foot washing but as much washing as Jesus wants to give him.
There is clearly an element here in which Jesus is foreshadowing that he is about to perform an even more lowly task for Peter (and the others, and us) that of dying on the cross for our sins.
Jesus then explains the message that he wants the disciples (and hence us) to take away which is that since he will perform not just the symbolic lowly act of foot washing, but also the ultimate lowly act of dying, for us, we should also serve one another.
When we as Christians re-perform the symbolic foot washing act we are reminding ourselves of this. However unlike Jesus our actions do not cleanse others from sin, so that part of the initial story is not part of the reenactment. Our actions merely teach and remind us of the call to mutual service and submission.