Texas and your home state of Washington are both members of Drivers License Compact (DLC) and Non-Resident Violator Compact. The DLC has its member states report information regarding moving violations that a driver received back to the person's home state. So the court in Texas will advise the Washington Department of Licensing of your citation.

The Non-Resident Violator Compact requires member states to suspend the driver’s license of those who get traffic tickets for moving violations in other states and fails to pay or otherwise take care of the ticket properly. Since WA and TX are both part of this it would appear that if you ignore the Texas speeding ticket that Washington State could suspend your license until you did take care of the TX ticket.

The Revised Code of Washington (RCW) Chapter 46.23 discusses Washington's participation in the Nonresident Violator Compact if you would like to read more about it there.

We know of no statute of limitations that you could wait out for the Texas ticket, it will likely stay with you until you take care of it. If you believe there is a limit of time in which Texas can try to collect on or enforce the ticket, you may want to check with a traffic ticket lawyer to get information on this.