You can be held legally responsible for necessary services (such as medical) provided to a spouse or a minor children (as well as possibly for a disabled adult child for which support has been ordered by the court). You would also be responsible for services that you have authorized and/or requested -- in much the same as if I take my neighbor's car to the mechanic and ask them to do work on it, the mechanic will come after me for the payment because I am the one who requested/authorized that the work be done (not my neighbor, though the mechanic could put a lien on the car).
I am not aware of any statute or legal theory that would make you financially responsible for the uncovered medical expenses (such as co-pays) of an adult child who is not disabled, regardless of who carries and pays for the medical insurance. I would assume it is not the "insured" portion of these services that the doctor is seeking payment on, but rather the uncovered/uninsured portion of such services.
The 19 year old is legally an adult and able to enter into binding legal contracts that can be enforced against him or her. If it is the 19 year old who is requesting and/or authorizing the medical services, then the medical services providers have a contract for payment for the serves with the 19 year old, not random family members who were not a party to the transaction.
It might not be a bad idea for you to contact any medical providers who have been attempting to collect money for these services from you and put them on notice that you are not and will not be financially responsible for any uncovered services that they may provide to the child. Then, when the child next goes to each of these providers, perhaps they will do a better job of demanding full payment of any uncovered services upfront, which may cut down on some of these medical bills that the child is "constantly creating".
You also can terminate them from your health insurance policy. However, if the child has any actual legitimate need for medical treatment/services on a fairly regular basis, then dropping the coverage for the child (for as long as the child remain eligible) may create more problems -- for the child, at least. The more hits they take to their credit report for unpaid medical bills, the more difficult it may be for them to actually get a job (since many employers do pre-screening credit checks these days).
There is one caveat and that would be if your child was the subject of a custody and support agreement with you and the child's other parent (which does not sound like the case based on your description). If there was a legally enforceable agreement or a court order that required you to provide insurance coverage and pay for some or all of any uncovered medical expenses until the child turned age "x" (21, 22, graduated college, turned 25, whatever), then you would have a legal obligation to comply with that order or agreement. However, even that would not be something that the doctor's office could try to enforce against you. They would go after your child and then the other parent would come after to you for your share of the payment in that scenario.
In any event, if the medical providers assert that you are financially responsible for the services, then I would suggest that you demand that they explain the legal theory on which they are basing their claim and then go consult with a lawyer in your local area (while a family law attorney may be able to assist you with this, you may want to try someone who does consumer rights/collections/contract law).
This response does not create an attorney-client relationship and is intended for general information purposes only.