My old iPhone is still receiving my texts?

Your old phone is still receiving iMessages. That's because you haven't removed it from your Apple account.

The new one is either a lost or stolen phone, or has been incorrectly reset by the last legitimate owner using Find My iPhone. So it's now expecting the last owner to enter their iCloud password and Apple username. If the last owner is unavailable to re-activate the phone then it's absolutely useless. An Apple Store can bypass it, but only if you can provide full ID and absolute proof that you're the legal owner. A fee is charged as well.

If the last owner re-activates it, then the correct way to reset the phone is for that person to go into Settings>General>Reset>Erase all data and settings. The last owner must enter his or her PIN to do that.

Once it's been wiped and reset the phone can be set up afresh on your Apple iCloud account.

Sign the old phone out of iMessage, and sign the new one in.