I need to transfer movies from an iPhone to a computer?

Okay, here goes. I synched my mother's iPhone 5 with my computer a year ago and added four movies I had made to the Videos app.
I have not synched it since then and now I no longer have access to the computer I synched it with. The hard drive the films were backed up on also failed, leaving the iPhone as the only source of these videos. Now I need the videos on my new computer and can't get them off!
Some details. My old computer was a 13-inch mid-2010 MacBook Pro. My new computer is a 15-inch late-2013 MacBook Pro.
The phone (as I previously mentioned) is an iPhone 5.
The checkbox "Manually manage music and videos" is checked.
I have tried dragging and dropping the videos from the "Movies" section.
I can't delete anything else off the phone like apps in order to get these movies.
And one last detail: I HATE iTunes because I'm always running into sync-related issues like this! I don't know if any of that helps, but someone please attempt to help!

Added (1). As I said, Logan, I no longer have access to the computer or my mother's "account". I'm not considering iCloud to come into the equation at all. It's too buggy and cumbersome right now to take seriously.

You can try this one: http://www.imobie.com/...honetrans/

Have you tried using iCloud as another alternative do it through Wi-Fi though as you may use a lot of data
Or simply log her account onto your laptop save movies on it and log off simple