I get no cold water from my single handed shower control.

I just rebuilt the single handle system here.It seems the mixer had developed some residue and the 'flutter valve' was not allowing the flow to occur.

Unless you are prepared to replace the handle,I would suggest not taking it apart yourself,this can be very tricky.I found the handle schematics online and saw what I needed to understand before I began.

I did this because a new handle system would cost somewhere around $400,I would rather try to fix something first.

Is this a new situation? IF so you most likely need to replace the cartridge. Do you know the brand name of your faucet?