Better to have easily restorable backups and just wipe & configure as a "new" device (not connected to cloud services) before entry. Restore from backup when you get home.
This is actually pretty painless these days. Sad that it's necessary.
If you want an Android smartphone that's really cheap go to Wal-Mart. Unlocked Android phones south of $60.
If you can do without a smartphone you can get a dumb or feature phone for ~$10-$30.
There might even be better deals on Aliexpress or somewhere similar if you shop around.
I would call neither _dirt cheap_ for _my_ budget, but that is what I had arrived at as the cheapest possible that is still usable too. "Dirt" cheap, not for me, or probably most Americans, but what can you do.
(FWIW, I think it's no longer possible to get a dumbphone that will work on AT&T for $10-$30, now that AT&T turned off 2G and requires 3G even for voice. More like $60-$80 at the very lowest end that will work on AT&T. Don't know about internationally. :( )
Incidentally, is this limited to phones? Are they only searching these accounts because they're authed to the phone they make you unlock? Or are they saying "oh, OK, so you're traveling without a phone. Please step over to this kiosk and log into your Facebook account" as well?