I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear bout your situation.
While your boyfriend unfortunately can ask you to leave his home by giving you a written notice at least 21 days before the end of the month, he has no legal right to either force you to leave your child behind or make you leave with no notice at all. RCW 59.18.200.
You can take your daughter with you when you move out and sue for full custody and child support. The default position is that a child benefits from having a relationship with both parents, but you should still be able to get support if you are the primary custodial parent, meaning that your daughter stays with you at least half the time. The judge will look at several factors to determine what is in the best interests of the child, including:
(i) The relative strength, nature, and stability of the child's relationship with each parent;
(ii) The agreements of the parties, provided they were entered into knowingly and voluntarily;
(iii) Each parent's past and potential for future performance of parenting functions as defined in *RCW 26.09.004(3), including whether a parent has taken greater responsibility for performing parenting functions relating to the daily needs of the child;
(iv) The emotional needs and developmental level of the child;
(v) The child's relationship with siblings and with other significant adults, as well as the child's involvement with his or her physical surroundings, school, or other significant activities;
(vi) The wishes of the parents and the wishes of a child who is sufficiently mature to express reasoned and independent preferences as to his or her residential schedule; and
(vii) Each parent's employment schedule, and shall make accommodations consistent with those schedules.
The primary determination will be the first factor. RCW 26.09.187. This calculator can help you figure out how much child support you might be able to get.
You also have the ability to file for an equitable distribution of assets acquired while the two of you were living together, on the basis that you have what is called a "meretricious relationship" - meaning a relationship that is like marriage. A judge is going to look at how long you lived together, whether you had a stable, committed relationship, whether you pooled assets, whether you were exclusive, whether you made employment decisions based on your relationship and similar factors to determine whether your relationship is considered marriage-like. This site has more information on that type of lawsuit.
These cases are difficult, so it may help to see if you can find a local attorney to help you out. That site above has some resources, or if there is a law school near you, it might help to reach out and see if they have a family law clinic that will help you.
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