Check the air in the bladder tank!
I'd check the air in the existing bladder tank. Then consider installing a Cycle Stop Valve or install a Pside-Kick www.cyclestopvalves.com somewhere between the pump and the tank. This will prevent your pump from cycling and give you constant pressure (like city water pressure) and extend the life of your pump, pressure switch and tank. A larger tank won't hurt but it won't help much either.
As far as the galvanized tank goes, if it isn't helping with anything leave it turned off or take it out. Personally I don't see the need of it.