The Part number is 4G7Z-9D290-AB It is not too expensive either. Remove the rear pan and then the mid pan. The sending unit is near the bottom of the oil filer assembly. You'll get about 1/2 cup of oil out when you remove the old one. Be ready with the new one to go in. I like to use teflon tape for sealing. Be careful not to over-tighten.
If gauge is moving "fluidly" but all over the place then best case is the sending unit. However, if the gauge needle is "sticky" then I'm afraid you may have a bad gauge. The sending unit is an inexpensive good try.
Just FYI, this is one of the parts readily available at local auto parts. Wells, Standard and Napa all repackage the same TRW switch that Motorcraft does in their own boxes. Likely others too. Its used on a number of cars, not GT specific.
I would also check the oil pump drive belt and pulleys since you have the two belles pans off. .Probably a sending unit, but others have had the pulley fail.. Most check this item as annual maintenance..