Can I use a Honeywell thermocouple in a State water heater?
First, I agree with the above comments:  I use a 400 fine-grit sandpaper to clean off the existing thermocouple tip and I clean off the pilot tip as well.  

As a separate precaution, I would check the spark on the igniter assembly by disconnecting it then watching for the small blue spark at the hand-button (DUH, only after turning off the gas!) 

Yes, you can use the Honeywell 30mV t-couple with the State if/when the threads in the gas control valve seat fit.

Lastly, note that some State water heaters have a high-temperature shunt soldered in-line with the thermocouple. These are a retrofit part to repair problems in hot-weather climates (like Arizona and Southern Nevada.)  In that case, the thermocouple with the shunt 'ignores' high ambient temperatures in garages, etc. I have had to replace my State water-heater thermocouple a few  times because the shunt resistor cracks. Of course the cracking problem only occurs in cold, winter temperatures. 

The parts list for your heater is found here; and it looks like yours (at least originally) did not have the high-temp shunt resistor version of the thermocouple.  Your thermocouple #9000056015 is manufactured by AO Smith and (for example) could be purchased here or elsewhere.

Bottom line:  go get the Honeywell and a 2nd push-button igniter; order a 2nd AO Smith thermocouple online; clean and test your existing ones if you are up for it....please post back your progress.