Recover deleted txt messages

Last night just after sending a text my phone died, I plugged it into my computer to charge. Once it turned back on I noticed my entire conversation was gone. I have called Apple and Verizon, as usual neither were helpful. I know that the messages are still hidden on the hard drive somewhere, but I need a way to access them.

Yes, I'm aware I can restore my phone with my iCloud account, but it's has been full for months.

Phone details, 5S running 7.1.2