While the escrow company should be liable, these cases can be a little messy.
The answer really depends on the details of what was written down in the escrow. While your oral statement to them is helpful, the court is first going to look at the written escrow instructions and distributions.
Generally, the escrow company's job is to get the payoff amount from all creditors (including the mortgage company) and then pay these amounts. (This is the primary function of an escrow agent/company).
If the escrow company fails to do this, they are in breach of both their contract to you, and they are negligent (their conduct falls below the standard of care for the industry.
You can sue the escrow company for damages in small claims court for up to $10,000.00 on both breach of contract and negligence causes of action (see: https://www.depts.ttu.edu/sls/forms/How-to-Sue-in-Small-Claims-Court.pdf)
Short of filing a lawsuit, you can try to mediate the dispute with them - contact your local bar association and request referrals to mediators, a third party neutral can often help you reach a mutually agreeable resolution. Use the bar association's referrals to contact a mediator or two, the mediator will then contact the other party to set up a mediation session, and you can go from there - hopefully resulting in a formal or written settlement agreement, and save yourself the time and expense of litigation.