That motor once started up will keep running no matter if the battery is having discharging issues. Age is age and 14 years are 14 years, worse if has sat in the water in a dock slip according to your post. Moisture will screw any electrical or elctronic part over time. What screws up any motor is not being run from time to time over time and where has been kept for all those years.
Check if someone can lend a Ignition Coil to swap and test while the combo runs on water at speed. If OK buy a new IC and save lots of cash by not taking the motor to nearest dealer. Would hate to pay $100 an hour for electrical testing..
Clean carb, check fuel pump cond if fuel starvation is suspect and make another test run at speed, if with no avail it's an electrical issue. A spark can jump properly when tested at neutral and 0 load, other scenario is jumping properly when the motor is at load say running midle to full throttle grip...
Happy Boating