The new wire seems to go to a junction box just a few feet away, then redirects outside. I'll take pics tonight and put them up so you can see the progression of the wire. I have a feeling I've bitten off more than I can choose.
Jim Port: Turns out my neighbor actually has done some work on this house (for the previous owner) and was able to come over and help me. The wires I thought were old and useless...were not (useless...they're definitely old) and I just needed to reconnect them. Simple fix...however he did mention some other pretty serious oversights near the washer/dryer that I need to take a look at. Thanks for all the suggestions and helping out a newbie!