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Every single Facebook user is or will soon be seeing a link at the top of their desktop news feed asking them to ”Stay in control of your account by following these simple security tips”. The link leads to the the Facebook Security page where users are taught how to spot a scam, pick a unique password, and most interestingly, confirm their mobile phone number for account recovery.
Facebook confirmed with me that the message will roll out to all desktop users over the next few days, and millions are seeing it right now.
I'm sure the black or gray market can supply what you need.
In the U.S. you can grab a prepaid phone at the local convenience store, it's just like a prepaid VISA. You use it until you run out of prepaid minutes and either replenish it or toss it when you're done, no questions asked.
So just make sure your face isn't visible to the convenience store security cameras when you buy the phone and you're good.