Well it's not that easy. Washington State has a toothless speedy trial rule. Each and every time your case gets continued by agreement, the time for trial starts over and the state has a fresh and new 90-day window in which to bring your case to trial. You do not need to sign a waiver for that to happen. The court rule dealing with speedy trial says it's automatic. And what is more, if the 90-day speedy trial period is exceeded, then the state's prosecutor gets an additional 30-days grace period. Getting a case dismissed on speedy trial ground in this state is not virtually impossible.
Im not sure why you would sign a speedy trial waiver at your arraignment, unless you wanted to have your next hearing past 90 days. I would speak with your public defender to see if that is what happened. Generally at arraignment the Judge will just toll your speedy trial rights until your next hearing if that is the case.
Regarding an argument to dismiss your case, more information would need to be provided. But as the other attorney stated, it is very difficult to get a case dismissed on a speedy trial violation in Washington these days. And there are all sorts of what are called "cure periods" where the Prosecutor can have additional time to bring your case to trial.
I would speak with your public defender and find out what happened at your arraignment.