I can't verify my e-mail address without changing my Apple ID? The iPhone keeps asking me to verify my e-mail address. How can I get around this? Thank you in advance

If you are the owner or the owner has shared all information with you then, in order to open the Activation Lock:- read here --> Turn off Activation Lock - Apple Support Activation Lock for iPhone, iPad, and iPod touch - Apple Support

If you have an unknown Apple ID on the App Store and Apple ID page. (See the pic below)

Learn how to remove a device from a previous owner's account.

If you are not the owner and do not have the Apple ID and Password then there is no way that you can get into this iPhone unless the original owner shares the Apple ID and password with you. If you are lucky you can get into this iPhone if they (owner) accidentally remove the device from their Apple ID.