The best course of action in the long run is to get a book like "the 12 volt doctor" or something similar explaining marine charging systems, then trace out and document your entire system since there's telling how the PO actually set it up.
Use a pencil, don't be afraid to erase. Once it is done you can color the lines red and black too, if that helps. Seriously.
In the meantime...Yes, using your Sears charger, set on the "slow" charge option should be just fine. At a 2A charge rate connected to two batteries (no need to split them) it almost certainly can't overcharge them in any short time. Check the electrolyte level first, top up with distilled water if needed, and use a cheap $20 multimeter to check the charge voltage. When the batteries reach a peak and then stop (somewhere between 12.8 and 13.6 volts) that's "full enough" for right now. Let them sit overnight, check the voltage again, if it has fallen below 12.4-12.6, give them another 24 hours of charge and call that good enough for now.
Most small outboards don't need a battery at all, once you add the electric starter they need power to start, and often have a small magneto to provide a little power while they are running, for the running lights and to recharge the starter drain.