Unable to send texts or phone calls

I have been unable to send any texts and unable to make phone calls. The messages app is very slow to load and sometimes my messages goes missing for a while. I have tried factory reseting it and it doesn't help at all. I have also gotten a new phone and there is no change. I believe that the reason it will not load is because I have to many existing messages in my app, but I can't delete them. I have tried countless times and I am just so frustrated and unsure of what to do. Is there a way that I can backup everything except my messages to iTunes? I would be more than willing to lose all my texts if it means it would work again. I would like to keep all my contacts but I am also willing to re enter them manually if need be. Any help would be much appreciated. I am almost ready to give up and I love Apple products but right now I am just so fed up with my phone.