My husband is in the military and has cheated on me several times and can I take him to court in Washington state for adultery?

You may report him for adultery to the military, The state seldom takes any of these cases any longer.

You go on to divorce court and move on with your life.

You cannot take him to court, I have never heard of this. I know of people who want to file civil suits, but they never get to the courthouse. There are simply too many adultery cases, it would take a whole separate part of the court to deal with these cases alone.
You can try to report him to the military for adultery, but trust me, speaking from experience, they will do absolutely nothing. Military & adultery almost go hand & hand, for many reasons. They'll play to you, saying they don't accept this type of behavior, etc, etc, etc. But honestly, they'll do nothing.
File for divorce based on adultery and move on. Don't waste your time trying to punish him... it will come back to him, it ALWAYS does.

When I was in the service, my commanding officer and sergeant major got calls from scorned spouses at least once a month. The CSM would call in the soldier in question and berate him for a while. Usually about getting caught.

While technically it is in violation of the Uniform Code of Military Justice, its not worth the efforts of the Army to prosecute it.

Like the others have said - file for divorce and move on.

Edited to add: if he is an officer they might take things a little more seriously - but the result will likely be the same.

In WA you can not take him to court for Adultry. It sounds like you want to hurt him. You should consider counceling and just file the paperwork. He doesn't sound like he is worthe the stress anyway. In the military they will not do anything trust me... I had tons of evidance for my ex and they did nothing. Im talking phone calls emails pictures you name it. Especially if he is in the Army...

First off, adultery and the military does not go hand in hand. People have low morals regardless of their occupation. The only way the military will pursue Uniform Code of Military Justice(UCMJ)punishment on your Soldier is if you have hard evidence. I.E. the Soldier in the act of committing or if the Soldier admits to this act. Pictures of them holding hands, eating dinner, etc. can be contested. If he/she admits to fact or if have hard evidence, take it to the Battalion Commander or Command Sergeant Major. They are obligated to address the issue and resolve the problem ASAP.
***In the end Just MOVE ON WITH YOUR LIFE***