As long as the nitrate does not bottom out, organisms find it far easier to consume nitrate than to consume sulfate, so they do not produce hydrogen sulfide.
Yes, if nitrate was completely depleted and there was organic matter remaining to degrade, hydrogen sulfide could be produced.
There is a well understood progression of reactions that I discuss here:
from it:
As mentioned above, this order of electron acceptors used to oxidize organic material is oxygen (O2), then nitrate (NO3-), then manganese (Mn++++), then iron (Fe++), then sulfate (SO4--). Researchers can plot the concentrations of these chemicals as a function of depth, and can also associate an
ORP with each transition, although some overlap of the chemistries takes place in each zone. The oxygen zone has an ORP of 0 to 600 mV, the nitrate zone is -150 to 550 mV, the manganese zone is -50 to 400 mV, the iron zone is -700 to -150 mV and the sulfate reduction zone is -850 to 0 mV. Consequently, if hydrogen sulfide is being formed from sulfate, the ORP is likely below 0 mv in that region.