My messages with only one person keeps disappearing

Only when the sender sends them; like most of the messages are gone. I am low on GB's on my phone but I don't understand why it has to affect my messages? Like nobody else's convos are missing but this one person's and it's very aggravating! I just want the messages back please help me! Also sometimes my phone acts like it's hesitant to send my texts as well, so I swipe off and come back and the sender and mine are gone! It's been doing this for a few days almost weeks now and I really need help with this! I'm thinking if I get a new phone will it help me out. Idk