This is the link with all the info you need.

https://www.dnr.wa.gov/programs-and-services/product-sales-and-leasing/firewood

Yes, in the national forest. It will depend on the area and time of year. Check out the website of the forest closest to you for more information.

From what i can understand you can’t cut trees yourself, you can only collect branches which aren’t really sustainable if you heat your home with wood

They aren't branches, I can tell you that. Our carport is full of rounds we pull from DNR sites. My brother in law has always heated his home with a wood stove from DNR wood. DNR permits, yes you are taking the wood that is left, not cutting it down.

As far as public lands to cut your own, actual trees, follow the other suggestion for potential on federal lands.

I am not sure, but I don't believe you can actually cut trees down on public land.

Some private timberland owners, such as Weyerhaeuser, and Green Diamond, sell access permits that may include firewood cutting.

DNR has its rules. USFS has separate rules: https://www.fs.usda.gov/detail/mbs/passes-permits/forestproducts/?cid=fseprd500137 The passes and precise rules are by forest (there are 7 separate national forests in Washington State.) You can only cut dead or downed trees, but there are lots of those out there. If you want to see who owns a particular area, you can click on any area on this recreation map and it will tell you the owner, the area name and give you a link. (You can download the free app if you want to navigate off-line) https://maps.northwestportal.com/outdoors