You are not measuring pH, you are measuring voltage. If the probe has a source resistance of 100M it is necessary to have an amplifier input resistance much above this to get a reasonable reading. Any standard voltage measuring device such as DMM or scope will have an input impedance of about 1M so will effectively short the input and you will measure nothing. The op-amp is there to sense the voltage and to provide an output with a lower output resistance so that a DMM can measure it. An oscilloscope is used to detect varying voltages, you should not have any such.
Any shunt resistance across the amp input will have a disastrous effect on the reading, in one example I saw (did you give the web site?) the amp was soldered directly to the BNC terminal so there was no printed circuit board or other components which could provide a track other than the package of the amp. If you have handled this with sweaty hands or tried to clean it, there may be sufficient leakage to affect the result.
Check your amp with a voltage source in series with the highest resistance you can find. The voltages should match, both positive and negative. Connect the probe and try in an acidic and alkaline solution, any difference? If so, do they agree with the data given by nbw?
Do to the very high source resistance and the cable capacitance it will take time for the readings to settle.
You say that the input measured 590mV but how could you do this without a very high input impedance device which is what you are tryiing to construct?