Dont worry about it. You will have cell phone records to show you called. I didnt have to use the automated, but I would call both. Not coming in this is Dilli and my employee ID is, Bye. They will not try to fire you based on their system that doesnt work, I promise. This is one area they will lose. you are protected, period. +1 for a govt agency.
If it is an emergency that came about at 8am, you have til 7pm and you have the 3 day no call no show. Relax, its not good to need fmla, if you got it, you need it, you have enough to deal with.