The new traffic tickets in Washington are now being printed out on a 8.5" by 11" sheet of paper with a landscape orientation. You do not need to sign the new tickets.
Now turning to the issues, as far as I can tell, Washington State Law is completely watered down on cross-jurisdictional exercise of police powers. I remember watching the old show Dukes of Hazard when the Dukes would lead the sheriff on a high speed chance to the county line and once across, the cops would pull off and throw his hat on the seat in disgust! Well partner, those days are a gone in this state.
Under the Washington Mutual Aid Peace Officers Powers Act, cities have the legal authority to contract with individual police officers from other jurisdictions to enforce city ordinances. Furthermore, under RCW 10.93.070, a general authority Washington "peace officer" has the legal authority to enforce traffic or criminal laws outside of their own jurisdiction in limited circumstances including:
(1) Upon the prior written consent of the sheriff or chief of police in whose primary territorial jurisdiction the exercise of the powers occurs;
(2) In response to an emergency involving an immediate threat to human life or property;
(3) In response to a request for assistance pursuant to a mutual law enforcement assistance agreement with the agency of primary territorial jurisdiction or in response to the request of a peace officer with enforcement authority;
(4) When the officer is transporting a prisoner;
(5) When the officer is executing an arrest warrant or search warrant; or
(6) When the officer is in fresh pursuit, as defined in RCW 10.93.120.
I don't know with New Castle and Renton has a mutual aid agreement, but they probably do. These are the in thing these days and many cities have these.
The other thing could be that the officer will say he was in acting "in response to an emergency involving an immediate threat to human life or property" or was "fresh pursuit" of you because you were in his opinion, violating the traffic laws.
Under Washington law, drivers now are not required to sign an infraction. Additionally, officers may pace drivers to determine their speed. To determine the likelihood of winning if you were to challenge your infraction, I strongly suggest meeting with an attorney that regularly handles traffic infractions.
Additionally, to preserve your rights, if you decide to "fight" the ticket, notify the relevant court (referenced on the infraction) that you intend to dispute the infraction within 15 days of its issuance.