MIKEMOL--- I am so sorry to hear about your messy experience. I know you must be really frustrated by now. It sounds that when you were running your engine, that you also had the fill plug removed at the same time (not good). Never, never, run the engine without the fill plug re-installed. In addition; never, never, remove the fill plug with engine running.
Yes, you can use the Syringe-Pump as a very convenient way to add AT fluid in a steady methodical manner and it will give you the ability to stop filling as soon as you encounter overflow out of the fill plug hole. Basically; with the engine off, fill with AT fluid until it overflows out of the fill hole. Then replace fill plug. Start engine, and cycle through each gear momentarily pausing in each gear for about 15 seconds to insure adequate cycling coverage of fluid. Then stop engine, and remove fill plug and refill until AT fluid again overflows through fill hole. Replace fill plug and repeat running engine through the gears. Stop engine and check fill hole and add more until overflow. When you get to the point when you go to remove the fill plug (with engine off) and AT fluid is already at the edge and you can't add any more AT fluid, then your entire system is completely full. Replace the fill plug with new gasket and torgue to final spec.
I would only use "regular" (not synthetic) DEXTRON III per the instructions. Do not vary from the explicit instructions. In addition, buy yourself a quality Torque Wrench as you should already have one in your tool box anyway. Take your fill plug bolt (as a sample) to NAPA or other quality auto parts store and ask them for an appropriate gasket (you'll need two). When you replace the two plugs with new gaskets for the final time, tighten with your new Torque Wrench to required spec. Do not take chances with factory Torque spec's in an effort to safe money by not buying a quality Torque Wrench. Those Torque spec's are put there for a reason, and you should follow them to the letter!!! Lastly; when replacing plugs, take special care not to accidentally "cross-thread" the plugs into the holes.