Expat marriage in Saudi Arabia

and I am unable to get through to anyone at my USA embassy

How is that even possible?  Go visit them, it is your embassy, you can pretty much go any time.  Check out the timings on their website.

Also, your country publishes detailed rules and policies on the internet regarding every issue concerning its citizens, including for US citizens wanting to marry non-US citizens!

If he is Muslim, the only way to get married here is to marry in a Shariah court.  If that's OK with you, do it.  If not, he has to go through the long procedure of applying for a fiance visa to the US (do a Google search).  Or, just take a flight to Mauritius and get married the day after you arrive.  All you need is passports and birth certificates!