i'm having the same problem. i only make new accounts to play games and deactivate them after a couple of hours (after i've done what i wanted to do of course). i made new accounts a few times a week maybe, if that ...
it use to be the phone verification would pop up randomly and it was no biggie, just give up on trying to create that account and create another instead.
these past couple of days though, the phone verification requests pops up on every account i tried to create. have been searching all last night and today for a solution but nothing yet. this site kept popping up on searches (funny that, lol) and looks the most hopeful of finding a solution in the future, so i joined.
though i've tried using proxies it's all a bit over my head and by the sounds of it, doesn't work with facebook anymore, anyway. it seems getting disposable cell phone numbers is the only solution at this point but they cost. i tried the 'pinger' site to get free disposable numbers but even if it did work (their servers are always down apparently), you'd have to keep signing up there for a new number for each facebook account you wanted.
so - i'm with you on this one TA and looking forward to any helpful suggestions/solutions.
eta: clearing cookies/cache, using cc cleaner and unplugging modem for a few hours doesn't work either.