Question
I have a couple of small plants that show lots of black specs near the base, is this normal, or a disease?

I think I've been keeping up with the watering, but the leaves are drying out and dying, slowly but surely.

Answer
Doesn't sound like a disease, but may be an insect.  Go to Home Depot and purchase Bayer Advanced Tree and Shrub Insecticide (they may have one named for indoor plants as well).  This product is mixed with water and poured around the base of the plant.  It is taken up systemically and will kill any bugs within a week or so.  It also provides protection from reinfestation for up to a year.  I hope this helps solve the problem.  Thanks.