You should not be listening to your husband on legal matters. Whether he is willing to sign court documents will not affect the court's ability to enter orders. If he does not want to take part in the court proceedings, the court will just enter the orders without his input.
Whether he wants to pay child support does not affect his legal duty to pay child support. If he is the father, the court will order him to pay child support as needed. Unless he refuses to go to work, the government will take some or all of the child support from his paycheck.
The court usually does not force parents to exercise their parental rights (such as spending time with the child). However, the court will force the parents to meet their parental duties (such as financially support their child).
If for some reasons you two do not want to be divorced, you can look into filing for legal separation. Legal separation is very similar to a dissolution of marriage (divorce) as far as child support and parenting plan are concerned. The major difference between legal separation and divorce is that the two spouses are still legally married with each other in a legal separation. Some couples want to remain still married with each other for medical insurance, religion, and other reasons.
What you should do is review your specific facts with your attorney. When both spouses agree and cooperate in the legal process, expenses and time can be reduced. However, one spouse cannot delay the legal process for very long if the other spouse wants to be divorced or legally separated.
The mandatory forms for most WA family law proceedings are free at http://www.courts.wa.gov/forms/ . From that site, there are links to instructions and self-help guides. The guides at WA LawHelp is informative.
There are family law facilitators who help people filling out family law forms. The facilitators charge no or low fees for their services. The facilitators usually have offices in the county's superior courthouse.
Excellent answer I would add:
The court process is not one that requires his cooperation. If he is properly served with a Summons and Petition and fails to respond a default can be entered against him. If he is in default, all of the relief requested in the Petition may be granted without further notice to him.
I often speak to parents that want to give up their rights to get out of child support. This is not possible, unless he is not the biological father to the child. Even if he is not the biological father, it may not be possible.
There is a presumption in favor of paternity for a child born during the marriage:
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