Correct, full branding for Writergrrls is Seattle Writergrrls. It is never Writer Grrls.
Always abbreviate as SW since SWG and WG are owned by Seattle Webgrrls and Writergrrls is not affiliated. Grrl(s) is also an accepted shortened form.
Capitalize "Writergrrl" when when making a general reference to someone who belongs to the organization.
EXAMPLE: Jane, a Writergrrl from Redmond, published her first short story.