Hope some other stitchers also reply with their advice as the products I use may not be available in your country.
I use Softly a woollen/delicate liquid to gently wash out the aida as I too have that grotty area from where the fabric covers the plastic frame. I think our palms sweat even after washing, so it cannot be helped, unless we wear gloves!
We have soaps too that are stain removing, (Sard, Preen) but great care has to be taken. And I usually use cold water only and rinse very well, swishing the fabric through running water. Much care has to be taken not to overrub/overscrub the dirty area too.
Here's some links to articles on this subject.