If we go with the theory that what we are seeing represents an open circuit gate input, and signals from hand capacitance are being coupled to it then there are at least 4 ways this could come about and we then would have to guess perhaps which one would be more likely:
1) there is a broken track or bad connection between IC's
2) there is a failing input connection inside an IC body
3) The output stage of an IC is going open circuit.
4) a complex failure inside the IC die.
In cases where an input is left floating, at least for TTL, it simply assumes a logic high, though it is less noise immune than if tied by a resistor to +5v. But, if there was a long length of track connected to it, it could pick up enough signal by stray coupling to activate the gate. Especially if the output stage of an IC driving an input was defective to the extent it kept it just near the gate's threshold voltage.
If a gate's input connection, to the pin on the IC package goes open circuit (I have seen this before a few times) the gate inside the IC acts like it is tied high. It would certainly have a very "short antenna" and not likely to be able to activate the gate, unless there was a more complex failure and some sort of cross coupling in the IC die.
So overall I think there must be an IC with a failing output stage, leaving a length of track connected to an IC's input pin partially open circuit condition and behaving like an antenna.
All TTL output pins should be able to drive the 30pF capacitance of a x10 scope probe, so I would look for any change in the fault, when the scope connects to an output pin on the IC body, and if it does, suspect that IC, being careful not to get confused where the presence of the hand and probe in the area causes a change in the fault. So I would go back and see what effect the scope has on output pin G5 pin 3, and if it is affecting that, suspect that IC and test all the other output pins on the other ICs with the scope.