Mysterious stains on white clothes after washing

I’ve noticed stains on my clothes after I wash them, before I put them in the dryer. They’re gray-looking, like oil stains of some sort. I’ve checked different blogs and they say it could be accumulation of liquid softener in the tub, or bearing and seals need to be replaced, or hoses need to be replaced. I don’t use liquid softener, so that’s not the problem. I’ve tried all sorts of stain removers, including Dawn, and nothing works. These are brand new clothes getting ruined from my 4-year-old Neptune washer. Can anyone help? Should I try and fix this myself or do I need to call a repair man. I’m very mechanically inclined so am not afraid to go at it.

Remove the rear access panel and check the bearing (behind the pulley).

Try to use Affresh, washer cleaner designed especially for front load washers.

Here are the breakdown diagrams and http://www.appliancepartspros.com/partsearch/model.aspx?diagram_id=629502&&model_id=47811

Gene.