Most verbal contracts are binding, although there are topics that require written contracts. But even if no written contract is required, it is not always easy to prove a contract was formed in the absence of a signed agreement. Then there is the issue of what are the terms of the contract. Since the written agreement was not signed, one or the other of the parties may object to any number of the key provisions as drafted. Your question is therefore one of facts and circumstances.
The short answer, as above, is yes. There are certain types of contracts - lengthy leases, marriage, sales of land, etc. - which require a written agreement. However, a general agreement to perform a function is enforceable under law.
The limitation period on oral contracts is the same 6 year period given to a written agreement. The fact that your oral agreement has been reduced to writing, though not signed, will assist in evidencing the obligations of the parties.
You may file suit to seek performance or damages from another party's breach. Consult an attorney to discuss your options in finding a suitable remedy.
Which statute of limitations should apply is sometimes in dispute. If you are concerned about the statute of limitations, you should check which statute would apply to your case.
RCW 4.16.080 (Actions limited to three years) provides: "The following actions shall be commenced within three years: ... (3) Except as provided in RCW 4.16.040(2), an action upon a contract or liability, express or implied, which is not in writing, and does not arise out of any written instrument;"