Good update @Justin R.At least you don't have a dumpster of stuff to clear out of there - you're already a step ahead.Don't forget to check the underside of the home for problems too.  

Buckets of moisture are migrating up out of the soil into your investment(lay a 2' x 2' piece of clear plastic on your lawn in the middle of summer and watch how quickly water droplets form on the underside). 

You might already know this, but: do the dishwasher drain hoses go up from the dishwasher, to either an air gap or the underside of the counter, before going back down to the sink drain or Disposall connection?

If there is access to underside of its floor and it's wood I would Try drilling and spraying a few cans of Great Stuff expanding foam thru the floor and see how that works. 

Adding acoustical insul between the floor joist to the under side of the floor works so.

You could drill a small hole on the underside and let any standing water drain, then when dry, you could put the cap on.

Have to insulate the attic in my duplex (frost forming on the underside of roof causing a leak). 

Are you guys talking about those planks with the fibrous underside to a thin laminate/printed surface?

In fact, there is so much air coming into the house as a whole that the furnace is constantly, constantly running - costing us a fortune just to keep it comfortable in here.I'm not sure how difficult it is to insulate the underside of the house (as well as the rest of the house).

depending on how bad it is and what you used to seal it in the first place, I would pull the drain apart and use 100% silicone on top and the underside of the drain assembly. once everything is tight wipe the extra off and you should be good as new.