I'm moving to WA from CA in January and I was wondering how much cash aid and food stamps would I get in that state its just me and my son.

More than likely you will have to legally live in Washington for a matter of months not days in order to qualify for any kind of financial aid. Best to look into that prior to moving. You may decide not to move. The usual "waiting" time is around 60 days if you have voluntarily left the other state to move to a new state. This is in place so folks don't just keep moving here and there to collect the cash benefits from the state. Also, your child needs to be under the age of 16 in order for you to get any kind of cash aid.

Department of Social and Health Services, DSHS, Washington State

Here is the site for Washington State. It does not state the residency time requirements but I know that there is some sort of residency time requirement - there always is.