Technically speaking, the statutory deferral procedure is only available for misdemeanors or for gross misdemeanors. Both of the offenses mentioned are infractions, subject only to a fine. You may be able to negotiate with a prosecutor to reach essentially the same end through a plea agreement.
As a last resort, the "no insurance" charge can be handled by way of community restitution -- community service hours performed at minimum wage in lieu of a monetary fine.
How to get the best result depends on what your primary concern is. If you are concerned about having the infraction convictions on your record, the best route will likely be with the prosecutor. If you are concerned with finances, the best route may be the community restitution program.
As an aside, you really should get at least minimum limits insurance coverage. It is both legally required and a very good idea to protect yourself from lawsuits.
You might be eligible to defer if you haven't had a ticket deferred within the last seven years (RCW 46.63.070). There are several really good attorneys in your area who primarily handle traffic tickets. Talk to a couple of them, and see if you are likely eligible to defer, or if some other option might be right for you. Most of them will offer a free consultation.
Yes, you can get a deferred finding for these tickets but beware! The Judge usually imposed the full fine for a deferred finding and can also order you to have no traffic violations for 12 months. The no insurance ticket alone is $550 and it is likely that expired tabs ticket you received was $124. Unless you have an insurance company like Progressive that increase your rate for no insurance tickets, these tickets are considered non-moving violations and most insurance companies will not increase your rates.
If you had insurance at the time of the ticket and file proof to the court, the Judge would likely dismiss the no insurance ticket and impose a $25 fine. If you obtain insurance after the time of the ticket and file proof to the court, the Judge would likely give you a substantial reduce of the fine depending on your record. That fine could be reduced to as low as $150.
I would consider how much you are willing to pay to have two non-moving violations cleared from record. You can only use one deferred finding every 7 years. I would advise not to use a deferred finding in this case unless you are facing other consequences or violations for a criminal charge like a DUI.