Greetings
Benasorus,
The apps that you mention seem to be free, so without resetting you phone, create a new gmail account for yourself. Make sure that your new account is set to synch. I think that there is a option for Google to back up your phone, (contacts, wifi settings and more) to its servers.
Connect to the play store, and Google should have a record of all the apps connected to this account.
There are a number of free apps that you can install to backup your sms and texts. You should be able for sent yourself any gmail messages from one account to the other.
The Verizon stuff should be tied to the phone account, and not your gmail account.
After your factory reset, input your new gmail account info, and you should be good to go.
As far as apps purchased under the school gmail account, I'm not totally sure about transferring between accounts. A Google search might give you some answers, but you may need to contact Google directly to have something done.
Hope this helps.
