CA refuses to sign agreement.

You cannot force a CA to sign anything....if it is a third party collection agency, as a general rule their clients set the terms and other than a settlement notice or paid in full notice, no other letters are permitted. Historically debtors have used payment plan letters to change their credit reports and to secure other credit...that is the major reason for the no payment plan letter rule.

You say you are trying to settle, and then you say repayment plan....you cannot do both.

I'd tell them that you're not paying anything until you have an agreement in writing. If they don't want to "sign" it, they can type up the agreement on their own letterhead and send it to you. That should work just as well.

Not all will do it. They will do settlement letters including 3 payments to settle, but the OP hasnt stated what type of terms they are after. Clients can be very picky.