You should see an IP attorney in the jurisdiction where the customer resides or in your locality. It will take an attorney to stop them, and it needs to be done promptly before they get too invested in your design to change.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
For legal and practical reasons, you have a weak case. From your post, you did the work with no contract or agreement on price. That not only potentially harms your case to sue for money, there was no agreement on the use of your work. You can still sue to get paid, but will have to show the value of your work since you can't rely on a contract price. Second, from a practical standpoint, a lawsuit requesting an injunction or restraining order will cost many thousands in legal fees. A suit to collect less than $15,000 can be filed in Magistrate Court. You might also get claims filed against you, as you state. You don't tell us what you believe to be the value of your work, but to some extent it is might be a valuable lesson learned about getting a firm agreement and getting paid upfront.
I would contact an attorney and get some advice on whether the amount owed justifies the expense of pursuing this matter legally. I think you should also seek advice on your potential liability. This sounds like something that may be worked out with help from an attorney. If you would like a free consultation, I will be happy to speak with you.
All the attorneys have given you good advice. I will only add that this is a public forum and you should be careful about discussing your case in any public forum as what you say will be used against you. Hire an attorney (call or email one from AVVO) to give you advice and to fight for your rights (which you have.)
My disclaimer is simply that Avvo already has an adequate disclaimer.