zinc-carbon cell

at anode, the zinc carbon cell body is made of zinc usually.
at cathode, the carbon rod is surrounded by MnO2
immediate inside the zinc body, in between with MnO2, there is a moist paste of ammonium chloride

so, for discharging, zinc would have reaction:
Zn2+ --> Zn2+ + 2e-
therefore, the zinc body becomes thinner and thinner, and eventually leakage occur
At the moisture paste, NH4+ <--> NH3 + H+ (equilibrium reaction)
the MnO2 is also possible for the reactions:
MnO2 + 2H+ + 2e- --> Mn2O3 + H2O
This add more moisture on the moisture paste at discharging.

thus, in some time, leakage occur so the moisture from the moist paste comes out...

less acidic, because MnO2 would trap H+ to form Mn2O3 and H2O

also, when you just put the cell alone without discharge,
Zn + H+ --> Zn2+ + H2(g)

therefore, the cell would eventually less acidic even you did not use it!