There are really two options here.
First, the simple option of just using restrooms on a different floor. I would recommend this approach frankly.
The fact that your management (at a Fortune 500 company!) isn't actively taking steps to address this is a bit weird to me, which strongly is suggesting something else is going on that you are unaware of - or your claims are greatly exaggerated here.
However, if you really want to fight this, your second option is this. You need to be very objective and delicate (particularly since you already tried and got shot down). I would recommend waiting a few weeks since you first complained to the management and then, if the problems continue, setting up time with those management in person and calmly raise the concerns of:
- The bathrooms are unsanitary to the point of unhealthiness
- It makes you uncomfortable working in an environment where there are significant health concerns
- Verify your management understands what actually are the problems
- You would like to know what steps are being taken to prevent the problems in the future
Come away from this meeting with action steps of some sort.
If you are not 100% confident in being able to do this in a respectful and polite way DO NOT DO THIS. Your post has the tone of someone who would not be good at the sort of nuance required for this, it might not be so, but this sort of approach will require a good deal of office politiking.