iPhone Texting issue

I have had a texting issue for several months. I have zero problems with texting other iPhones using iMessage (whether in groups or individually). However, I have significant limitations when texting non-iPhones. I can send and receive simple texts to a single person. But, if I try texting a picture to somebody I am already texting with, the image won't go through. Further, I can receive group texts from groups of other non-iPhone users, but if I try to reply to the group, the text will not go through. Or, if I even try to start a new group text that includes non-iPhone users, the text will not go through. (And when I say group, even if it's just 2 other people the text will not work.)

This issue first arose on my old iPhone 5s while I was with AT&T, however, the issue has followed me to my new iPhone 8 on Verizon (so the issue is not with the provider or the hardware). As I backed up my new iPhone 8 from my old iPhone 5s, I'm guessing it's something with the settings on my phone that got copied over from the old phone.

Under my "Messages" setting, "Send as SMS" is turned on and "MMS Messaging" and "Group Messaging" are also both turned on. I have turned the phone off and on. I have no pending updates from my carrier settings. I have reset my network settings, and none of those have done anything for me.