Wow, creepy. There are multiple layers of concern I would have here. I agree with previous replies about the apple ID being the likely culprit for her being able to text as your kids. I agree with a factory reset.
If she had her phone and laptop linked (mirroring text messages, emails, phone calls on the laptop in case the phone is somewhere else in the house), part of this behavior may simply be due to continued mirroring if she accidentally or "accidentally" forgot to sign out of her apple ID and iCloud accounts on the phone before gifting it. You should be able to make a first stab at a fix by going to Settings > Phone > Calls on other devices - uncheck everything. Then go to Settings > Messages and delete everything from Text Message Forwarding. If iMessage is activated, click on send and receive and uncheck everything except for the device phone number. If you're still seeing texts go through at that point, turn off iMessage and let whomever she is contacting know to block the sender of that message string and only respond to texts from the actual phone number.
Resetting the phone to factory settings is the ultimate solution, but may be problematic if still signed into her apple ID and/or iCloud, as it requires the apple ID password to log out of iCloud or perform a reset (so that you can't steal a phone from someone's hands while they are using it and it is unlocked, hit factory reset and then sell it as a blank phone). It's worth a shot, though.
How to erase your iPhone, iPad, or iPod touch
If that doesn't work, and she's willing to be the good guy, she can log into her iCloud account online and delete that device from her apple ID and Find My iPhone. If she does that, you should know it, as the phone will prompt you for a new apple ID and you should be able to perform the factory reset as above.
If this fails, I would go to a genius bar in an apple store. If you explain the creepy aspect of the situation, they might be able to help you (but may have their hands tied by your non-ownership of the phone). They are generally very good, particularly if you ask a question that the first level person can't answer and get to one of the more senior people. One question to ask regardless is whether this phone is compatible with your current cell provider, as at least older iPhones are typically "locked" to either CDMA or GSM networks unless specifically purchased as unlocked.
If you are able to get it reset, you're still likely connecting to the network using the Dick Cheney phone plan for kids, AKA Verizon Smart Family. Think creepy and intrusive, and you'll be on the right track.
Verizon Smart Family FAQs - Set parental controls
I would definitely switch it onto your plan as an extra line (assuming that they haven't activated Number Transfer Freeze with Verizon). If you are on Verizon and get it switched onto your account, log into your verizon account immediately and activate Number Transfer Freeze for all lines as well as Enhanced Authentication (two-factor authentication that goes to your phone anytime you try to log in to your verizon account or speak with customer service.) I have Verizon and didn't know about these until just now - I plan to activate them.
If you hit a dead end at any step in the above progression, a trip to the post office to mail it back is probably your best option.
That's really subtle and, I imagine, pretty unnerving to discover. Sucks. Sorry it happened to you.
