In Washington state, legally separated filing jointly, can one spouse living in the house qualify the couple for capital gains exemption when selling the home?

If only one spouse meets the "use test" (living in the home for 2 out of the last five years before the sale), you can exclude up to $250,000 of capital gain.

Only one spouse has to own the home, but both spouses have to live in the home for two years to qualify for the full $500,000 exclusion.

See Publication 523.

When filing jointly, you would claim either $250,000 if one spouse qualifies, or $500,000 if both spouses qualify.   The spouse who moves out qualifies for their $250,000 even if they did not live in the home 2 of the past 5 years  as long as part of the separation agreement allowed the other spouse to remain in the home.   This is on page 3 of publication 523, special rules for divorce and separation.