It may not be a problem with incoming water. If you have a front loading washer, check for an accumulation of muck in the folds of the door seal. A top loader can also have a pocket of retained soil.
As for the lavatory, is there a slow drip at the faucet? I'd clean it with a 25% bleach solution and see if it reoccurs.

It may be difficult, but try to determine whether the 'black slim' is organic or inorganic.

Black manganese and some types of sulfur may leave similar residue.

I have tested many city water applications where no chlorine was reaching the homes. This can be due to a couple of reasons: the distance from the water plant is too far; the chlorine (or chloramine) injections are too low; or another element is inhibiting the municipal disinfection system.

If that is so, then orgnic deposits are contributing to your dilemma.

Hot water tanks can be a breeding place for bacteria 9none that I know are pathological) or it could be a case of mold, whic can be harmful.

Bottom-line, determine the cause before trying too many solutions.

Andy Christensen, CWS-II

Thanks for the suggestions.
As a dyed-in-the-wool DIYer, I am mostly reluctant to
call for professional help.
Do you think it is time to pay someone to analyze the water?
Most of the ads for these companies indicate that they
solve problems with well systems, not city water.

How do I determine if it is organic problem?
I have cleaned everything thoroughly.
Can I test the water myself?

Thanks
John