This may sound silly, or maybe you've already tried it, but, try signing out of your Apple ID (Settings>iTunes & App Store>, click on Apple ID, on pop-up click "Sign Out"), do a power cycle (not sleep mode), then come back here and sign in, try again.
I also read again how Apple states AirDrop works, perhaps you do need to be on the same wifi network, not sure.
Have you sent the others your contact info? Try giving at least one person your contact info via text/imessage, have them add it to their contacts. I say this because, even though the other had my contact info already, this worked.
A shot in the dark: are you able to iMessage with them ok? If not, perhaps you need to turn iMessage off/on to reset the settings.
If nothing helps, a trip to the Apple store if you're close, perhaps there is some hardware issue with your 5s?
I don't need any sort of contact info (as far as I know) because I'm using the "Everyone" option of AirDrop, not "Contacts Only". I can iMessage the two phones fine (well actually just one of them because I don't even have the other iPhone's number).
I'm guessing the Apple store is the best bet, although I really want to know why this is happening.
AirDrop drops to everyone as I see it, no option to single out one specific user...so for me airdrop is useless.
I would be afraid to use it once since this appears to be the way it works.
Tap on one user to send the file only to them...