This helped me with my original question and for all of those who were wondering. My Apple ID is one of my emails. I had this email in my contact and was also using it for FaceTime on my iPad. I followed the steps below and things changed instantly without restart:
1. I went into my contacts and pulled up my own contact. Took out my email address and changed it with another one I use. When I did that my wife's phone posted our iPad for her picture.
2. Went to the iPad contact in my phone and changed it to another email address. (We have our iPad as a contact for iMessage and FaceTime directly to the iPad) When I did that, it did not change her name and picture..still said iPad.
3. When into her phone and changed the iPad contact information. When I looked on my phone, FINALLY, my wife's name and picture was showing as the airdrop contact.
It does seem to be a glitch with the system for those who use an email address as their Apple ID user name. So if you have a family, run through the contacts and take out the email section if it is the SAME as the Apple ID username. Again, I started with my phone, because it was the phone presenting my name on the other devices. So, if your daughters name, son's name, wife's name, mom or dad's name is showing as the airdrop name and pic for your device then start with that person's device. Chances are in the contacts somewhere the email is listed for the contact pushing their names and pictures to the other devices for airdrop's feature.
Again, I have one Apple ID for all of my devices. I did not change the Apple ID as not to hinder grabbing apps from the cloud on all devices for my family, sharing calendars, notes, etc. All I had to do was remove the email in my contact because it was the same as the Apple ID. I also suppose the email can be changed for the Apple ID too if that is easier. This was the only thing that worked for me and I hope it helps everyone else out. Credit to Hane Hormes for fixing this. I did not think of it before him. I appreciate your help with this!!