I do not believe it is possible to purge child support records. If you had a founded allegation through CPS - ie.. a finding after an investigation - there may be a way to litigate it or over turn it - however, you will need to determine if that is what exists and likely prove you had no notification of it - in order to contest it. You need to consult with an attorney ion a more lengthy setting than the internet.
I do not practice in Washington but have been keeping an eye on the issue you raise. You should talk with a CPS defense attorney to find out your statutory rights there and look at the articles I have linked below to see if you may need a broader challenge to Washington's CPS registries to get the relief you seek.
If you can prove the you were never notified of the original finding by CPS and your right to a due process hearing about the validity of the "charge", you have a better chance of undoing it than if there was a finding that you ignored at the time. The fact that you never had to go to court is irrelevant. CPS can conduct their investigations and make findings without ever taking a case to court.
A CPS finding that you have neglected your own children, unless it was life threatening neglect, should not interfere with a nursing career, though you may have to fight through a lot of bureaucratic red tape at each stage of your education, licensing and job placement process to have the 15 year old record be appropriately ignored.
NOTE: This answer is made available by the lawyer for educational purposes only. By using or participating in this site you understand that there is no attorney client privilege between you and the attorney responding. This site should not be used as a substitute for competent legal advice from a licensed professional attorney in your area who regularly practices in the subject matter which your question is about. You should develop an attorney client relationship with the lawyer of your choice so that your communications will be subject to the attorney client privilege and have the other benefits of a professional relationship. The law changes frequently and varies from jurisdiction to jurisdiction. The information and materials provided are general in nature, and may not apply to a specific matter as partially described in the question.
Sign up to receive a 5-part series of useful information and advice about child custody law.