Vinegar and baking soda are a time tested drain opener that you've apprently already tried. If that's only lasting a few days, you might try repeated back-to-back applications. If that doesn't
work, I suspect you have a grease buildup in your
galley (kitchen) sink and most likely soap scum, tooth paste etc buildup in the
head (bathroom) sink. Repeaed applications of undiluted Dawn Ultra dishwashing liquid should clean out both and won't upset your gray water tank.
You don't say what part of the world you're in, but if in the US, it's
legal to send gray water directly
overboard in all US waters except for a few inland lakes--Lake George in
New York, Lake Winnipeksaukee in New Hampshire and at least one small on in
California (name escapes me at the moment). Unless you're on one of them, you might consider rerouting your sink drains to an above-waterline thru-hull....after you've pumped out and thoroughly rinsed out your gray water tank.
--Peggie